744 (number): Difference between revisions

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Cardinal	seven hundred forty-four
Ordinal	744th; (seven hundred forty-fourth)
Factorization	23 × 3 × 31
Divisors	1, 2, 3, 4, 6, 8, 12, 24, 31, 62, 93, 124, 186, 248, 372, 744
Greek numeral	ΨΜΔ´
Roman numeral	DCCXLIV
Binary	10111010002
Ternary	10001203
Senary	32406
Octal	13508
Duodecimal	52012
Hexadecimal	2E816

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744 (seven hundred [and] forty four) is the natural number following 743 and preceding 745.

744 plays a major role within moonshine theory of sporadic groups, in context of the classification of finite simple groups.

Number theory

744 is the nineteenth number of the form $pqr^{3}$ where $r$ , $p$ and $q$ represent distinct prime numbers (2, 3, and 31; respectively).^[1]

It can be represented as the sum of nonconsecutive factorials $k!$ ,^[2] as the sum of four consecutive primes $p$ ,^[3] and as the product of sums of divisors $\sigma (n)$ of consecutive integers $n$ ;^[4] respectively:^[a]

{\begin{aligned}744&=4!+6!\\744&=179+181+191+193\\744&=\sigma (15)\times \sigma (16)=24\times 31\\\end{aligned}}

744 contains sixteen total divisors — fourteen aside from its two unitary divisors — all of which collectively generate an integer arithmetic mean of $120=5!$ ^[9]^[10] that is also the first number of the form $pqr^{3}.$ ^[1]^[b]

The number partitions of the square of seven (49) into prime parts is 744,^[14] as is the number of partitions of 48 into at most four distinct parts.^[15]^[c]

It is palindromic in septenary (2112₇), while in binary it is a pernicious number,^[19] as its digit representation (1011101000₂) contains a prime count (5) of ones.^[d]

744 is abundant^[24] and semiperfect,^[25] as well as practical.^[26] It is the first number to be the sum of nine cubes in eight or more ways.^[27]^[e] In decimal, 744 is the number of six-digit perfect powers.^[29]

Totients

744 has two hundred and forty integers that are relatively prime or coprime with and up to itself, equivalently its Euler totient.^[5]

This totient of 744 is regular like its sum-of-divisors, where 744 sets the twenty-ninth record for $\sigma (n)$ of 1920.^[30]^[f] Both the totient and sum-of-divisors values of 744 contain the same set of distinct prime factors (2, 3, and 5),^[32] while the Carmichael function or reduced totient (which counts the least common multiple of order of elements in a multiplicative group of integers modulo $n$ ) at seven hundred forty-four is equal to $\lambda (744)=30=2\times 3\times 5$ .^[33]^[g] 744 is also a Zumkeller number whose divisors can be partitioned into two disjoint sets with equal sum (960).^[37]^[h]

Of these 240 totatives, 110 are strictly composite totatives that nearly match the sequence of composite numbers up to 744 that are congruent to $\pm {1}{\bmod {6}}$ , which is the same congruence that all prime numbers greater than 3 hold.^[39]^[i] Only seven numbers present in this sequence are not totatives of 744; they are 713,^[j] 589,^[k] 527,^[l] 403,^[m] 341,^[n] 217,^[o] and 155;^[p] all of which are divisible by the eleventh prime number 31. The remaining 130 totatives are 1 and all the primes between 5 and 743 except for 31 (all prime numbers less than 744 that are not part of its prime factorization) where its largest prime totative of 743 has a prime index of 132 (the smallest digit-reassembly number in decimal).^[70]^[q] On the other hand, only three numbers hold a totient of 744; they are 1119, 1492, and 2238.^[5]^[r]

744 is the sixth number $n$ whose totient value has a sum of divisors equal to $n$ : $\sigma (\varphi (744))=744$ .^[73] A total of seven numbers that have sums of divisors equal to 744, they are 240, 350, 366, 368, 575, 671, and 743.^[52] If only the fourteen proper divisors of 744 are considered, then the sum generated by these is 1175, whose six divisors contain an arithmetic mean of 248,^[10]^[s] the third-largest divisor of 744. Otherwise, the aliquot sum of 744, which represents the sum of all divisors of 744 aside from itself, is 1176^[11] which is the forty-eighth triangular number,^[6] and the binomial coefficient $\operatorname {C(49,2)}$ present inside the forty-ninth row of Pascal's triangle.^[74]^[t] Only one number has an aliquot sum that is 744, it is 456.^[11]^[u]

Graph theory

The number of Euler tours (which are Eulerian trails of directed graphs) of a complete graph $K_{6}$ on six vertices and fifteen edges is 744.^[82] On seven vertices, there are 129,976,320 Euler tours. These can only be generated on complete graphs with at least three vertices; the number of tours for three, four and five vertices are respectively 2, 2 and 264 (the latter is the second digit-reassembly number in base ten).^[70]^[v]

Otherwise, 745 is the number of disconnected simple labeled graphs covering six vertices, where the most symmetric of these graphs has three pairs of distinct vertices each covered by one edge alone and all three edges intersecting; this yields the disconnected covering graph $\{\{1,4\},\{2,5\},\{3,6\}\}$ on vertices labelled $1$ through $6$ in a hexagonal arrangement, with the remaining 744 graphs representing all other possible conformations.^[84]

Convolution of Fibonacci numbers

744 is the twelfth self-convolution of Fibonacci numbers, which is equivalently the number of elements in all subsets of $\{1,2,3,...,11\}$ with no consecutive integers.^[85]^[86]^[87]

Abstract algebra

The j–invariant holds as a Fourier series $q$ –expansion,^[α]

j(\tau )=q^{-1}+744+196\,884q+21\,493\,760q^{2}+864\,299\,970q^{3}+\cdots

The friendly giant $\mathrm {F_{1}}$ contains an infinitely graded faithful dimensional representation equivalent to the $q$ coefficients of this series, where $q=e^{2\pi i\tau }$ and $\tau$ the half-period ratio of an elliptic function.^[92]

Also, the almost integer^[93]

e^{\pi {\sqrt {163}}}\approx 262\,537\,412\,640\,768\,743.999\,999\,999\,999\,250\,072\,59\approx 640\,320^{3}+744.

This number is known as Ramanujan's constant, which is transcendental.^[94] Mark Ronan and other prominent mathematicians have noted that the appearance of $163$ in this number is relevant within moonshine theory, where one hundred and sixty-three is the largest of nine Heegner numbers that are square-free positive integers $d$ such that the imaginary quadratic field ${\textstyle \mathbb {Q} \left[{\sqrt {-d}}\right]}$ has class number of $1$ (equivalently, the ring of integers over the same algebraic number field have unique factorization).^[95]^{: pp.227, 228} $\mathrm {F_{1}}$ has one hundred and ninety-four (194) conjugacy classes generated from its character table that collectively produces the same number of elliptic moonshine functions which are not all linearly independent; only one hundred and sixty-three are entirely independent of one another.^[96] The linear term of error $O$ for Ramajunan's constant is approximately,

{\frac {-196\,884}{e^{\pi {\sqrt {163}}}}}\approx {\frac {-196\,884}{640\,320^{3}+744}}\approx -0.000\,000\,000\,000\,75,

where $196\,884$ is the value of the minimal faithful complex dimensional representation of $\mathrm {F_{1}}$ , the largest sporadic group.^[97]

Specifically, all three common prime factors $(2,3,5)$ that divide the Euler totient, sum-of-divisors, and reduced totient of $744$ are the smallest and only primes that divide the orders of all twenty-six sporadic groups, in contrast with only six groups $(\mathrm {F_{1}} ,{\text{ }}\mathrm {B} ,{\text{ }}\mathrm {F_{3}} ,{\text{ }}\mathrm {Ly} ,{\text{ }}\mathrm {ON} ,{\text{ }}\mathrm {J_{4}} )$ whose orders can be divided by the largest consecutive supersingular prime and largest prime factor of seven hundred and forty-four, $31$ ;^[95]^{: pp.244–246} three of these belong inside the small family of six pariah groups that are not subquotients of $\mathrm {F_{1}} .$ ^[98] The largest supersingular prime that divides the order of $\mathrm {F_{1}}$ is $71$ ,^[99]^[100] which is the eighth self-convolution of Fibonacci numbers, where $744$ is the twelfth.^[86]^[β]

The largest three Heegner numbers with $d>19$ also give rise to almost integers of the form $e^{\pi {\sqrt {d}}}$ which involve $744$ . In increasing orders of approximation,^[109]^{: p.20–23}^[γ]

{\begin{aligned}e^{\pi {\sqrt {19}}}&\approx {\color {white}000\,0}96^{3}+744-0.22\\e^{\pi {\sqrt {43}}}&\approx {\color {white}000\,}960^{3}+744-0.000\,22\\e^{\pi {\sqrt {67}}}&\approx {\color {white}00}5\,280^{3}+744-0.000\,0013\\e^{\pi {\sqrt {163}}}&\approx 640\,320^{3}+744-0.000\,000\,000\,000\,75\end{aligned}}

Square-free positive integers over the negated imaginary quadratic field with class number of $2$ also produce almost integers for values of $d$ , where for instance there is $199\,148\,648^{2}-0.000\,97\ldots \approx e^{\pi {\sqrt {148}}}+(8\times 10^{3}+744).$ ^[111]^[112]^[δ]

E₈ and the Leech lattice

Within finite simple groups of Lie type, exceptional Lie algebra ${\mathfrak {e_{8}}}$ holds a minimal faithful representation in two hundred and forty-eight dimensions, where $248$ divides $744$ thrice over.^[121]^[122]^: p.4 John McKay noted an intersection between finite simple groups of Lie type and those that are sporadic, where symmetries of nodes in the Dynkin diagrams of complex Lie algebra ${\mathfrak {e}}_{8}$ as well as those of ${\mathfrak {e}}_{7}$ and ${\mathfrak {e}}_{6}$ respectively coincide with the three largest conjugacy classes of $\mathrm {F_{1}}$ ; where also the corresponding McKay–Thompson series $j(\tau )^{1/3}$ of sporadic Thompson group $\mathrm {Th}$ holds coefficients representative of its faithful dimensional representation — also minimal at $\operatorname {dim} 248$ ^[123]^[97] — whose values themselves embed irreducible representation of ${\mathfrak {e_{8}}}$ .^[124]^: p.6 In turn, exceptional Lie algebra ${\mathfrak {e_{8}}}$ is shown to have a graded dimension $j(q)^{1/3}$ ^[125] whose character $\chi$ lends to a direct sum equivalent to,^[124]^{: p.7, 9–11}

\chi _{e_{8}\oplus e_{8}\oplus e_{8}}(q)=J(q)+744=j(q),

where the CFT probabilistic partition function for

\mathrm {F_{1}}

is

J(q)

of character

\chi _{F_{1}}.

^[126]

The twenty-four dimensional Leech lattice $\Lambda _{24}$ in turn can be constructed using three copies of the associated $\mathbb {E_{8}}$ lattice^[127]^[128]^{: pp.233–235}^[ε] and with the eight-dimensional octonions $\mathbb {O}$ (see also, Freudenthal magic square),^[133] where the automorphism group of $\mathbb {O}$ is the smallest exceptional lie algebra ${\mathfrak {g_{2}}}$ , which embeds inside ${\mathfrak {e_{8}}}$ . In the form of a vertex operator algebra, the Leech lattice VOA is the first aside from $V_{1}$ (as $\mathbb {C} _{24}$ ) with a central charge $c$ of $24$ , out of a total seventy-one such modular invariant conformal field theories of holomorphic VOAs of weight one.^[134] Known as Schellekens' list, these algebras form deep holes in $V_{\Lambda _{24}}$ whose corresponding orbifold constructions are isomorphic to the moonshine module $V_{2}$ ^♮ of $\mathrm {F_{1}}$ ;^[135] of these, the second and third largest contain affine structures $E_{8,1}^{3}$ and $D_{16,1}E_{8,1}$ that are realized in $\operatorname {dim} 744$ .^[ζ]^[140]^[141]

Other properties

744 is also the sum of consecutive pentagonal numbers $P_{n}$ ,^[142]^[143]

744=P_{11}+P_{12}+P_{13}=210+247+287.

It is also the magic constant of a six by six magic square consisting of thirty-six consecutive prime numbers, between 41 and 223 inclusive (respectively, the thirteenth and forty-eighth prime numbers).^[144]^[y]

There are 744 ways in-which fourteen squares of different sizes alone fit edge-to-edge inside a larger rectangle.^[147]

Notes

Higher arithmetic

^ 744 is equal to the sum between the 41st and 44th indexed prime numbers, inclusive. Indices 15 and 16 of σ(n) multiply to 240, which is the Euler totient value of 744,^[5] and add to 31 which is the largest prime that is not a totative of 744 (less than).
A sum between 24 and 31 generates the tenth triangular number 55, where the eleventh triangular number is 66.^[6] 121 is the sum between these two triangular numbers, which is equivalent to the square of 11. The prime index of 31 and its permutable prime in decimal (13) form the third pair of twin primes $(11, 13)$ ,^[7] whose sum is 24, with respective prime indices 5 and 6^[8] that add to 11.
^ 120 is also equal to the sum of the first fifteen integers, or fifteenth triangular number $\textstyle {\sum _{i=1}^{15}i}.$ ^[6] This value is also equal to the sum of all the prime numbers less than 31 that are not factors of 744 except for 5, and including 1. Inclusive of 5, this sum is equal to $125 = 5 3$ , which is the second number after 32 to have an aliquot sum of 31.^[11]
In the Collatz conjecture, 744 and 120 both require fifteen steps to reach 5, before cycling through {16, 8, 4, 2, 1} in five steps.^[12]^[13] Otherwise, they both require twenty steps to reach 1 or nineteen steps to reach 2, which is the middle node in the {1,4,2,1,4...} elementary trajectory for 1 when cycling back to itself.
^ The radical $186 = 2 \times 3 \times 31$ of 744 has an arithmetic mean of divisors equal to 48,^[10] where the sum between the three distinct prime factors of 744 is $62 = 36$ . 186 is nontotient^[16] and noncototient,^[17] equal to the number of pentahexes when rotations are counted as distinct.^[18]
^ It is the four hundred and sixth indexed member, where 406 is the twenty-eighth triangular number;^[6] in its base-two representation, the digit positions of zeroes are in $1:1:3$ or $3:1:1$ ratio with the positions of ones, which are in $1:3:1$ ratio. Its ones' complement is 100010111₂, equivalent to $279 = 3 2 \times 31$ in decimal, which represents the sum of GCDs of parts in all partitions of $16 = 4 2$ .^[20] It is also the number of partitions of $62 = 2 \times 31$ (a divisor of 744) as well as 63 into factorial parts (without including 0!),^[21] and the number of integer partitions of 44 whose length is equal to the LCM of all parts^[22] (with 63 the forty-fourth composite number, where 44 is itself the number of derangements of 5).^[23]
^ On the other hand, every positive integer is at most the sum of two hundred and seventy-nine eighth powers (Waring's problem), preceded by a maximum number of ${1, 4, 9, 19, 37, 73, 143}$ $n$ -powers;^[28] where 279 is the decimal representation of the ones' complement of 744 in binary. In this sequence, the first six members generate a sum equivalent to the seventh member 143, which is the sum of seven consecutive primes starting from 11 through the eleventh prime number: $11 + 13 + 17 + 19 + 23 + 29 + 31$ . It is also the product between the third twin prime pair $(11 \times 13)$ ;^[7] while figuring in $34 + 4 4 + 5 4 + 6 4 = 7 4 - 143$ , which is the first exception to the pattern of polynomials that starts with $32 + 4 2 = 5 2$ and $33 + 4 3 + 5 3 = 6 3$ .
^ The value of this sigma function represents the fifteenth sum of non-triangular numbers in-between triangular numbers; in this instance it is the sum that lies in-between the fifteenth (120) and sixteenth (136) triangular numbers^[31] (i.e. the sum of $121 + 122 + ... + 135$ ).
^ The thirtieth triangular number is $465 = 3 \times 5 \times 31$ , equal to the difference between 744 and 279 (equivalently 1011101000₂ and 100010111₂, ones' complement pairs respectively). 30 is the twelfth number $m$ such that $6 m + 1$ and $6 m - 1$ are twin primes (181, 179).^[7]^[34] It is also the binomial $(31,2)$ equal to the number of size-2 subsets of ${0, 1, ..., 31}$ that contain no consecutive integers (in light as a triangular number, trailing its sequence by one index).^[35] In the Padovan sequence, 465 is the twenty-eighth indexed member equal to the number of compositions of 28 into parts congruent to $2 mod 3$ , also the number of compositions of 28 into parts that are odd and greater than or equal to 3, and the number of maximal cliques in a $(28 + 6)$ –path complement graph (specifically, in the $34$ –path).^[36]
^
The two sets are:
- (1, 2, 3, 4, 8, 12, 62, 124, 744)
- (6, 24, 31, 186, 93, 248, 372)
960 is also the sum of six consecutive prime numbers 149 + 151 + 157 + 163 + 167 + 173, between the 35th and 40th primes (it is the thirty-fifth such number).^[38] The fifteenth and sixteenth triangular numbers generate the sum 120 + 136 = 256 = 2⁸ that is the totient value of 960.^[5]
^ The composite totatives are
{25, 35, 49, 55, 65, 77, 85, 91, 95, 115, 119, 121, 125, 133, 143, 145, 161, 169, 175, 185, 187, 203, 205, 209, 215, 221, 235, 245, 247, 253, 259, 265, 275, 287, 289, 295, 299, 301, 305, 319, 323, 325, 329, 335, 343, 355, 361, 365, 371, 377, 385, 391, 395, 407, 413, 415, 425, 427, 437, 445, 451, 455, 469, 473, 475, 481, 485, 493, 497, 505, 511, 515, 517, 529, 533, 535, 539, 545, 551, 553, 559, 565, 575, 581, 583, 595, 605, 611, 623, 625, 629, 635, 637, 649, 655, 665, 667, 671, 679, 685, 689, 695, 697, 703, 707, 715, 721, 725, 731, 737}.
Its smallest composite totative is $52 = 25$ , that is the only number to have an aliquot sum equal to 6;^[11] where the twenty-fifth prime number 97 is generated from a sum of 48 and 49. This is the first odd prime number that is not a cluster prime^[40] (which precedes the thirty-first prime number 127 in this sequence), where the prime gap for a cluster prime is six or less;^[41] the sixth and seventh primes that are not cluster primes are the forty-eighth and forty-ninth primes 223 and 227. 97 is also the eleventh prime number $p$ of the form $6 m + 1$ for a natural number $m$ .^[42]
^ 713 is the product of the ninth and eleventh prime numbers $23 \times 31$ , where the twenty-third prime number (83) is the eleventh prime of the form $6 m - 1$ ; it is equal to the difference between 744 and 31, with an aliquot sum of 55^[11] and totient of 660.^[5]
^ ∗ 589 equal to $19 \times 31$ is the sum of three consecutive primes (193 + 197 + 199).^[43] It is also the ninth centered tetrahedral number, where 5 and 15 are the first two such numbers ( $121 = 11 2$ is the fifth).^[44] It is the fourteenth third spoke of a hexagonal spiral,^[45] and the twentieth quasi-Carmichael number,^[46] with a reduced totient of 90.^[33] In the spt function, 589 is the total number of smallest parts (counted with multiplicity) in all partitions of 15.^[47]
^ ∗ 527 equal to $17 \times 31$ is the number of partitions of 31 with equal number of even and odd parts.^[48] Its aliquot sum is 49^[11] whose arithmetic mean of divisors is $122 = 144$ , with a sum of its prime factors equal to 48. It is also the maximal number of pieces that can be obtained by cutting an annulus with 31 cuts,^[49] and equivalent to the sum of thirty-one consecutive non-zero integers $2 + 3 + ... + 32$ ; the second-smallest such sum after the third perfect number 496.^[50]
^ ∗ 403 equal to $13 \times 31$ is the thirty-seventh number to return 0 for the Mertens function,^[51] and the value of the sum-of-divisors of 144.^[52]
^ ∗ 341 equal to $11 \times 31$ is the sum between seven consecutive primes $(37 + 41 + 43 + 47 + 53 + 59 + 61)$ , of prime indexes 12 through 18. It is the smallest Fermat pseudoprime to binary,^[53] and the sixth centered cube number.^[54] It is the number of partitions of 31 such that every part occurs with the same multiplicity,^[55] and the thirty-first number in the Moser–de Bruijn sequence to be a sum of distinct powers of 4 ( $44 + 4 3 + 4 2 + 4 1 + 4 0$ ).^[56]^[57] 341 is also the eleventh octagonal number,^[58] equal to the number of nodes in a regular eleven-sided undecagon with all diagonals drawn.^[59] $s (341) = 43$ ,^[11] with an arithmetic mean of divisors equal to 96.^[10] It is also the tenth Jacobsthal number,^[60] and the total number of largest parts (or equivalently sum of smallest parts) in all partitions of 16.^[61]
^ ∗ 217 equal to $7 \times 31$ is the ninth centered hexagonal number, which also includes as previous members {7, 19, 37, 61, 91, 127, 169}^[62] where 127 is the sixth member that is also the thirty-first prime number. 217 is also the sixth non-trivial dodecagonal number^[63] and third non-trivial centered 36-gonal number.^[64] It is the third Fermat pseudoprime to quinary after 124 (a divisor of 744) and 4,^[65] and the fifteenth Blum integer whose distinct prime factors are congruent to $3 mod 4$ .^[66]
^ ✶ 155 equal to $5 \times 31$ has an aliquot sum of 37^[11] and an arithmetic mean of divisors equal to 48.^[5] It is the third number equal to the sum from its lowest prime factor through its largest, the only smaller numbers with the same property are 10 and 39^[67] which generate a sum of $49 = 7 2$ ; the next such number is 371, 1 less than half of 744. There are one hundred and fifty-five total primitive permutation groups $(G, X)$ of degree $81 = 9 2$ that only preserve partitions on the set $X$ by $G$ that are trivial;^[68] the sum of the number of all permutation groups of lower degrees is 666, which is doubly triangular,^[69] since it represents the sum of the first thirty-six integers, where 36 is itself the eighth triangular number.^[6] The one hundred and fifty-fifth indexed practical number is 744.^[26]
^ These prime totatives generate a sum of 44,647 while the composite totatives collectively generate a sum of 44,632; these sums have a difference of 15.
On the other hand, differences and sums between the six numbers congruent $\pm1 mod 6$ that are not totatives of 744 are themselves equivalent to divisors of 744, since they are biprimes in proportion with 31 (for example $713 - 589 = 124$ and $713 - 527 = 186$ ), while also generating other relevant equalities such as $713 - 217 = 496$ .
The greatest difference between the largest (713) and smallest (155) of these totatives is 558, the seventh number such that the sum of largest prime factors of numbers from 1 to 558 is divisible by 558, the previous and sixth indexed number is 62,^[71] the tenth-largest divisor of 744, whose sum of divisors is equal to 96;^[52] the fifth number in this sequence is 32 (which divides 96).
The difference between 558 and 62 is also 496, equal to $713 - 155 - 62$ .
^ The difference between 2238 and 1492 is 746, where 2238 is twice 1119. Specifically, $746 = 2 \times 373 = 1 5 + 2 4 + 3 6 = 2! + 4! + 6!$ is nontotient,^[16] and equal to the number of non-normal orthomagic squares with sum of entries equal to 6.^[72] Its totient value of 372 is half of 744.^[5]
^ 248 is twice 124 and half 496; the latter which is the third perfect number^[50] — like 6 and 28 (twice 14) — that is also the thirty-first triangular number,^[6] with form $2 p - 1 (2 p - 1)$ and $p = 5$ by the Euclid-Euler theorem. The totient of 248 is 120,^[5] with eight divisors that produce an integer arithmetic mean of 60;^[10] 496 has a totient of 240 like 744.
Where the sum between 744 and the totient $256 = 2 8$ of 960 (the Zumkeller half of 744) is equal to $1000 = 10 3$ , their difference is equivalent with $488 = 240 + 248.$
^ 1176 is also one of two middle terms in the twelfth row of a $\operatorname {(2,3)} -$ Pascal triangle.^[75] In the triangle of Narayana numbers, 1176 appears as the fortieth and forty-second terms in the eighth row,^[76] which also includes 336 (the totient of 1176)^[5] and 36 (the square of 6). Inside the triangle of Lah numbers of the form $\textstyle {L(n,k)=\left\lfloor {n \atop k}\right\rfloor }$ , 1176 is a member with $n = 8$ and $k = 6$ .^[77] It is a self-Fibonacci number; the fifty-first indexed member where in its case $F_{1176}$ divides $F_{F_{1176}}$ ,^[78] and the forty-first 6-almost prime that is divisible by exactly six primes with multiplicity.^[79]
^ 456 is the sixth icosahedral number,^[80] that includes 124 as the fourth indexed member in this sequence, which is the eleventh non-unitary divisor of 744 that is equal to the sum of all the prime numbers that are not part of its prime factorization, less than 31 (i.e. $5 + 7 + ... + 29$ ). It is the fourth perfectly partitioned number after 1, 2 and 3.^[81]
^ Half of 264 is 132, the number of irreducible trees with fifteen vertices.^[83]
^
71 is the twentieth prime number and 31 the eleventh. In turn, 20 is the eleventh composite number^[101] that is also the sixth self-convolution of Fibonacci numbers before 38, which is the prime index of 163.
71 is also part of the largest pair of Brown numbers (71, 7), of only three such pairs; where in its case 7² − 1 = 5040.^[102]^[103] Consequently, both 5040 and 5041 can be represented as sums of non-consecutive factorials, following 746, 745, and 744;^[2] where 5040 + 5041 = 10081 holds an aliquot sum of 611, which is the composite index of 744.^[101]
5040 is the nineteenth superabundant number^[104] that is also the largest factorial that is a highly composite number,^[105] and the largest of twenty-seven numbers n for which the inequality σ(n) ≥ e^γnloglogn holds, where γ is the Euler–Mascheroni constant; this inequality is shown to fail for all larger numbers iff the Riemann hypothesis is true (known as Robin's theorem).^[106] 5040 generates a sum-of-divisors 19344 = 13 × 31 × 48 that itself contains four divisors in proportion with 744 (and therefore, divisors also in proportion with 248 as well); which makes it one of only two numbers out of these twenty-seven integers n in Robin's theorem to hold σ(n) such that 744m | σ(n) for any subset of divisors m of n; the only other such number is 240:^[107]
- $19344 \div 26 = 744$
- $9672 \div 13 = 744$
- $1488 \div 2 = 744$
Where also 19344 ÷ 78 = 248, with 248 and 744 respectively as the 24th and 30th largest divisors (where in between these is 403 = 13 × 31, which is the middle indexed composite number congruent ±1 mod 6 less than 744 that is not part of its composite totatives); 2418, the 35th largest, is the seventh number n after 744 such that σ(φ(n)) is n.^[73] Furthermore, in this sequence of integers in Robin's theorem, between 240 and 5040 lie four numbers, where the sum between the first three of these 360 + 720 + 840 = 1920 is in equivalence with σ(744). The first number to be divisible by all positive non-zero integers less than 11 is the penultimate number in this sequence 2520, where 2520 − 840 − 720 = 960 represents a Zumkeller half from the set of divisors of 744,^[37] with σ(720) = 2418^[10] (and while 720 + 24 = 744 = 6! + 24). 5040 = 7! = 10 × 9 × 8 × 7 is divisible by the first twelve non-zero integers, except for 11.
^ Otherwise, for almost integers with nearness $|nint(x) - x | \leq 10 -2$ , the smallest such number is $e π \sqrt 6$ with $n$ of 6, approximately equal to 2197.99087.^[116] Whereas 2198 holds eight divisors that produce an arithmetic mean of 474^[10] (and where 474 holds a sum-of-divisors equal to 960,^[52] also the Zumkeller half from the set of divisors of 744),^[37] 2199 is the sixteenth perfect totient number,^[117] with an aliquot sum of 737^[11] equivalent with the largest composite totative of 744. Also, $2197 = 13 3$ .
There are a total of twenty-six almost integers with $|nint(x) - x | \leq 10 -2$ where $x \equiv e π \sqrt n$ and $n \leq 10 3$ , where the largest $n$ holds a value of 986.^[116] The sum between the upper and lower bounds $n$ less than one thousand with this almost integer degree of nearness is equal to $6 + 986 = 992 = 248 \times 4$ .
Also, the difference between the largest class $1$ and $2$ square-free integers is $427 - 163 = 264$ . Regarding square-free integers of class number $h = 3$ , there are a total of sixteen (or twenty-five when including non-maximal orders),^[118] the largest with value of 907 that is the 155th indexed prime number;^[8] of these only 59 produces an almost integer of the form $x \equiv e π \sqrt n$ with $|nint(x) - x | \leq 10 -2$ . In the full list of largest square-free positive integers with class numbers $h \leq 100$ ,^[119]^[120] ten of these maximum values are held by larger integers when non-maximal orders in their respective classes are included, where nine of these are uniquely biprimes divisible by 163 (the only exception is the largest value for square-free integers of class number 16); the largest of these in this bound is $821683 = 71 2 \times 163$ of class number $h = 70$ .^[109]^: 18−20
^ The magic square is:
${\begin{bmatrix}139&113&151&131&83&127\\223&149&89&47&157&79\\173&103&181&167&59&61\\67&137&53&97&211&179\\101&199&73&109&71&191\\41&43&197&193&163&107\end{bmatrix}}$
This is the second-smallest magic constant for a $6 \times 6$ magic square consisting of thirty-six consecutive prime numbers, the smallest such constant is $484 = 22 2$ ^[145] whose aliquot sum of 447 is the reverse permutation of the digits of 744 in decimal.^[11]
An $11 \times 11$ magic square that is normal has a magic constant of 671,^[146] which is the sixth number to have a sum-of-divisors equal to 744.^[52]

Heegner numbers, E₈ and the Leech lattice

^ The $j$ –invariant can be computed using Eisenstein series $E 4$ and $E 6$ , such that:
$j(𝜏) = 1728 .mw-parser-output .sfrac{white-space:nowrap}.mw-parser-output .sfrac.tion,.mw-parser-output .sfrac .tion{display:inline-block;vertical-align:-0.5em;font-size:85%;text-align:center}.mw-parser-output .sfrac .num{display:block;line-height:1em;margin:0.0em 0.1em;border-bottom:1px solid}.mw-parser-output .sfrac .den{display:block;line-height:1em;margin:0.1em 0.1em}.mw-parser-output .sr-only{border:0;clip:rect(0,0,0,0);clip-path:polygon(0px 0px,0px 0px,0px 0px);height:1px;margin:-1px;overflow:hidden;padding:0;position:absolute;width:1px}E4(𝜏)3/E4(𝜏)3 − E6(𝜏)2 ,$
where $E 4 (𝜏) = 1 + 240 \sum \infty n =1 (n 3 q n / 1 - q n)$ and $E 6 (𝜏) = 1 - 504 \sum \infty n =1 (n 5 q n / 1 - q n)$ , with $q = exp(2π i 𝜏)$ .
The respective $q$ –expansions of these two Eisenstein series have coefficients whose numerical values are in proportion to 240 and −504, respectively;^[88]^[89] where specifically the sum and difference between the absolute values of these two numbers is $240 + 504 = 744$ and $504 - 240 = 264$ .
Furthermore, when considering the only smaller even (here, non-modular) series $E 2 (𝜏) = 1 - 24 \sum \infty n =1 (nq n / 1 - q n)$ , the sum between the absolute value of its constant multiplicative term (24) and that of $E 4 (𝜏)$ (240) is equal to 264 as well. The 16th coefficient in the expansion of $E 2 (𝜏)$ is −744, as is its 25th coefficient.^[90]
Alternatively, the j–invariant can be computed using a sextic polynomial as: $j (λ) = 2 8 \times (λ 2 - λ + 1) 3 / λ 2 (λ - 2) 2$ where $λ$ represents the λ−modular function, with $256 = 2 8$ .^[91]
^ The sequence of self-convoluted Fibonacci numbers starts {0, 0, 1, 2, 5, 10, 20, 38, 71, 130, 235, 420, 744, 1308, 2285, 3970, 6865, 11822...}. The sum of the first seven terms (from the zeroth through the sixth term) is equal to 38, which is equivalent to the $a(n) = 7$ member in this sequence. Taking the sum of the three terms that lie between 71^[w] and 744 (i.e. 130, 235, 420) yields 785, whose aliquot sum is 163.^[11] 785 is the 60th number to return $0$ for the Mertens function, which also includes 163, the 13th such number. 785 is also the number of irreducible planted trees (of root vertex having degree one) with six leaves of two colors.^[108]
^ 163 is the thirty-eighth indexed prime and 67 the nineteenth (with the biprime $38 = 2 \times 19)$ , for values $d$ . In the approximation for the almost integer containing the fourteenth prime number as the seventh Heegner number 43 for $d$ , 960 is equivalent to the sum of either of two sets of divisors of 744 that collectively add to $1920$ , as mentioned in § Totients. Likewise, in the approximation for the almost integer containing the eighth Heegner number 67 for $d$ , 5280 is equal to the sum between 240 and 5040, which are the only two numbers in the set of twenty-seven integers in Robin's theorem for the Riemann hypothesis that have a set of divisors divisible by 744;^[107] 5280 also lies between the 131st pair of twin primes $(5281, 5279)$ ,^[7] respectively the 701st and 700th prime numbers, where 5281 is the 126th super-prime.^[110]
^ In the list of thirteen integers $n \leq 1000$ that yield almost integers — with nearness $|nint(x) - x | \leq 10 -3$ for values $x \equiv e π \sqrt n$ — the smallest $n$ of these is 25, that is also the smallest composite totative of 744. Less than 10³, the largest such $n$ is $719 = 6! - 1$ , where 719 represents the 128th indexed prime number;^[8] the sum generated by these two $n$ is in equivalence with $719 + 25 = 744.$
The middle indexed value in between these two bounds is 148, the twelfth square-free positive integer $d$ over the negated imaginary quadratic field of class number 2, followed by 163 and 232, the latter of which is the fourteenth square-free positive integer $d$ over the imaginary quadratic field $\sqrt- d$ of class number 2 (only these two numbers 148 and 232 for $d$ in this field of class number 2 yield almost integers with $|nint(x) - x | \leq 10 -3$ , where 163 is the largest over the same field with class number 1).^[112] In this same list, the three distinct non-supersingular primes (37, 43, 67) that only divide orders of pariah groups are also $n$ (less than 148) that yield almost integers of the form $e π \sqrt n$ .
In the almost integer representation for $e π \sqrt 148$ specified, $8 \times 10 3 = j (\sqrt(2i)) = 20 3$ , which is the class $1$ j−invariant of order $d = -8$ ,^[113] that is also a root in $Ψ 2 (X, X) = -(X - 8000) \times (X + 3375) 2 \times (X - 1728)$ , a polynomial for supersingular 2-isogeny graphs with loops (it is also a root in $Ψ 3$ ).^[114]^[115] The remaining two $n$ (53, 74) in the list of these almost integers with nearness $|nint(x) - x | \leq 10 -3$ of $n$ less than 148 generate a sum of $58 + 74 = 132$ equal to the prime index of 743,^[8] the largest prime totative of 744 (and, where $2 \times 37 = 74 = 148 \div 2$ ).^[x]
^ The 𝔼₈ lattice holds two hundred and forty root vectors that are represented by the vertex arrangement of the $421$ polytope, whose Petrie polygon is a thirty-sided triacontagon,^[129]^[130] where the numerical value of the Carmichael function $λ (n)$ at seven hundred and forty-four is 30,^[33] which is the Coxeter number $h$ of Coxeter group $E 8$ . Also, Coxeter numbers of simple reflexions in $E 6$ and $E 7$ are of orders 12 and 18, together equivalent to the Coxeter number of $E 8$ ;^[128]^: 234 where $E 6$ and $E 7$ are embedded inside $E 8$ . Four-dimensional $H 4$ hexadecachoric symmetry also contains a Coxeter number of 30, where $H 4$ is the higher-dimensional analogue of icosahedral symmetry $I h$ .
This 𝔼₈ lattice structure can be constructed with one hundred and twenty quaternionic unit icosians that form the vertices of the four-dimensional 600-cell, whose symmetries are rooted in three-dimensional icosahedral symmetry $I h$ of order 120;^[131] a value equal to the total number of reflections of Coxeter group $E 8$ ,^[128]^{: 226–232} that is equal to the arithmetic mean of divisors of 744.^[10] In total, a regular icosahedron and dodecahedron contain thirty-one axes of symmetry; six five-fold, ten three-fold, and fifteen two-fold — a count equal to the largest prime factor of 744.^[132] 240, the number of root vectors of $𝔼 8$ , is also the number of integers that are relatively prime with and up to 744,^[5] and the sum-of-divisors of this number is equivalent to $σ(240) = 744$ .^[122]^: p.21
^ $V E ‍ 3 8$ is isomorphic to the tensor product $V E 8 \otimes V E 8 \otimes V E 8$ ; also, affine structure $D 16$ is different from $D 16 +$ , which is associated with even positive definite unimodular lattice $𝔻 16 +$ .^[136]^[137]
$E ‍ 3 8,1$ and $D 16,1 E 8,1$ are associated with codes $e ‍ 3 8$ and $d 16 e 8$ that are two of only nine in-equivalent doubly even self-dual codes of length 24 and weight 4.^[138]^[139] The largest of these VOAs $V D 24$ is realized in $dim 1128$ , where successively halving its dimensional space leads to a 70½–dimensional space.

References

^ ^a ^b Sloane, N. J. A. (ed.). "Sequence A189975 (Numbers with prime factorization pqr^3.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-07-16.
^ ^a ^b Sloane, N. J. A. (ed.). "Sequence A060112 (Sums of nonconsecutive factorial numbers.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-07-16.
^ Sloane, N. J. A. (ed.). "Sequence A034963 (Sums of four consecutive primes.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-07-16.
^ Sloane, N. J. A. (ed.). "Sequence A083539 (a(n) is sigma(n) * sigma(n+1) as the product of sigma-values for consecutive integers.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-07-16.
^ ^a ^b ^c ^d ^e ^f ^g ^h ⁱ ^j Sloane, N. J. A. (ed.). "Sequence A000010 (Euler totient function phi(n): count numbers less than or equal to n and prime to n.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-07-16.
^ ^a ^b ^c ^d ^e ^f Sloane, N. J. A. (ed.). "Sequence A000217 (Triangular number: a(n) is the binomial(n+1,2) equivalent to n*(n+1)/2 that is 0 + 1 + 2 + ... + n.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-09-02.
^ ^a ^b ^c ^d Sloane, N. J. A. (ed.). "Sequence A001359 (Lesser of twin primes.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-10-18.
^ ^a ^b ^c ^d Sloane, N. J. A. (ed.). "Sequence A000040 (The prime numbers.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-09-06.
^ Sloane, N. J. A. (ed.). "Sequence A003601 (Numbers n such that the average of the divisors of n is an integer: sigma_0(n) divides sigma_1(n).)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-07-16.
^ ^a ^b ^c ^d ^e ^f ^g ^h Sloane, N. J. A. (ed.). "Sequence A102187 (Arithmetic means of divisors of arithmetic numbers (arithmetic numbers, A003601, are those for which the average of the divisors is an integer).)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-07-16.
^ ^a ^b ^c ^d ^e ^f ^g ^h ⁱ ^j ^k Sloane, N. J. A. (ed.). "Sequence A001065 (Sum of proper divisors (or aliquot parts) of n: sum of divisors of n that are less than n.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-09-02.
^ Sloane, N. J. A. (ed.). "3x+1 problem". The On-Line Encyclopedia of Integer Sequences. The OEIS Foundation. Retrieved 2023-09-22.
^ Sloane, N. J. A. (ed.). "Sequence A006577 (Number of halving and tripling steps to reach 1 in '3x+1' problem, or -1 if 1 is never reached.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-09-22.
^ Sloane, N. J. A. (ed.). "Sequence A000607 (Number of partitions of n into prime parts.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-07-16.
^ Sloane, N. J. A. (ed.). "Sequence A347586 (Number of partitions of n into at most 4 distinct parts.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-07-16.
^ ^a ^b Sloane, N. J. A. (ed.). "Sequence A005277 (Nontotients: even numbers k such that phi(m) equal to k has no solution.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-08-27.
^ Sloane, N. J. A. (ed.). "Sequence A005278 (Noncototients: numbers k such that x - phi(x) equal to k has no solution.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-09-24.
^ Sloane, N. J. A. (ed.). "Sequence A001207 (Number of fixed hexagonal polyominoes with n cells.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-09-24.
^ Sloane, N. J. A. (ed.). "Sequence A052294 (Pernicious numbers: numbers with a prime number of 1's in their binary expansion.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-08-28.
^ Sloane, N. J. A. (ed.). "Sequence A078392 (Sum of GCD's of parts in all partitions of n.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-10-01.
^ Sloane, N. J. A. (ed.). "Sequence A064986 (Number of partitions of n into factorial parts (0! not allowed).)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-10-01.
^ Sloane, N. J. A. (ed.). "Sequence A316433 (Number of integer partitions of n whose length is equal to the LCM of all parts.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-10-01.
^ Sloane, N. J. A. (ed.). "Sequence A000166 (Subfactorial or rencontres numbers, or derangements: number of permutations of n elements with no fixed points.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-10-01.
^ Sloane, N. J. A. (ed.). "Sequence A005101 (Abundant numbers (sum of divisors of m exceeds 2m).)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-04-03.
744 is the 181st indexed abundant number.
^ Sloane, N. J. A. (ed.). "Sequence A005835 (Pseudoperfect (or semiperfect) numbers n: some subset of the proper divisors of n sums to n.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-04-03.
744 is the 183rd semiperfect number.
^ ^a ^b Sloane, N. J. A. (ed.). "Sequence A005153 (Practical numbers)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-04-03.
^ Sloane, N. J. A. (ed.). "Sequence A345547 (Numbers that are the sum of nine cubes in eight or more ways.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-07-16.
^ Sloane, N. J. A. (ed.). "Sequence A002804 ((Presumed) solution to Waring's problem: g(n) equal to 2^n + floor((3/2)^n) - 2.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-10-01.
^ Sloane, N. J. A. (ed.). "Sequence A075308 (Number of n-digit perfect powers.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-07-16.
^ Sloane, N. J. A. (ed.). "Sequence A034885 (Record values of sigma(n).)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-07-16.
^ Sloane, N. J. A. (ed.). "Sequence A006002 (a(n) equal to n*(n+1)^2/2 (Sum of the nontriangular numbers between successive triangular numbers. 1, (2), 3, (4, 5), 6, (7, 8, 9), 10, (11, 12, 13, 14), 15, ...))". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-08-27.
^ Sloane, N. J. A. (ed.). "Sequence A081377 (Numbers n such that the set of prime divisors of phi(n) is equal to the set of prime divisors of sigma(n).)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-07-16.
^ ^a ^b ^c Sloane, N. J. A. (ed.). "Sequence A002322 (Reduced totient function psi(n): least k such that x^k is congruent 1 (mod n) for all x prime to n; also known as the Carmichael lambda function (exponent of unit group mod n); also called the universal exponent of n.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-08-27.
^ Sloane, N. J. A. (ed.). "Sequence A002822 (Numbers m such that 6m-1, 6m+1 are twin primes.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-10-02.
^ Sloane, N. J. A. (ed.). "Sequence A161680 (a(n) is the binomial(n,2): number of size-2 subsets of {0,1,...,n} that contain no consecutive integers.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-10-02.
^ Sloane, N. J. A. (ed.). "Sequence A000931 (Padovan sequence (or Padovan numbers)...)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-10-02.
^ ^a ^b ^c Sloane, N. J. A. (ed.). "Sequence A083207 (Zumkeller or integer-perfect numbers: numbers n whose divisors can be partitioned into two disjoint sets with equal sum.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-07-16.
^ Sloane, N. J. A. (ed.). "Sequence A127333 (Numbers that are the sum of 6 consecutive primes.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-08-27.
^ Sloane, N. J. A. (ed.). "Sequence A038509 (Composite numbers congruent to +-1 mod 6.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
^ Sloane, N. J. A. (ed.). "Sequence A038133 (From a subtractive Goldbach conjecture: odd primes that are not cluster primes.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-09-26.
^ Blecksmith, Richard; Erdős, Paul; Selfridge, J. L. (1999). "Cluster Primes". The American Mathematical Monthly. 106 (1): 43. doi:10.2307/2589585. JSTOR 2589585. Zbl 0985.11041.
^ Sloane, N. J. A. (ed.). "Sequence A002476 (Primes of the form 6m + 1.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-09-26.
^ Sloane, N. J. A. (ed.). "Sequence A034961 (Sums of three consecutive primes.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-09-24.
^ Sloane, N. J. A. (ed.). "Sequence A005894 (Centered tetrahedral numbers.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-09-24.
^ Sloane, N. J. A. (ed.). "Sequence A056107 (Third spoke of a hexagonal spiral.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-09-24.
^ Sloane, N. J. A. (ed.). "Sequence A257750 (Quasi-Carmichael numbers.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-09-24.
^ Sloane, N. J. A. (ed.). "Sequence A092269 (Spt function: total number of smallest parts (counted with multiplicity) in all partitions of n.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-09-24.
^ Sloane, N. J. A. (ed.). "Sequence A045931 (Number of partitions of n with equal number of even and odd parts.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-09-24.
^ Sloane, N. J. A. (ed.). "Sequence A000096 (a(n) equal to n*(n+3)/2.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-09-24.
^ ^a ^b Sloane, N. J. A. (ed.). "Sequence A000396 (Perfect numbers k: k is equal to the sum of the proper divisors of k.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-09-24.
^ Sloane, N. J. A. (ed.). "Sequence A028442 (Numbers k such that Mertens's function M(k) (A002321) is zero.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-09-24.
^ ^a ^b ^c ^d ^e Sloane, N. J. A. (ed.). "Sequence A000203 (a(n) equal to sigma(n), the sum of the divisors of n. Also called sigma_1(n).)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-09-11.
^ Sloane, N. J. A. (ed.). "Sequence A001567 (Fermat pseudoprimes to base 2, also called Sarrus numbers or Poulet numbers.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-09-28.
^ Sloane, N. J. A. (ed.). "Sequence A005898 (Centered cube numbers: n^3 + (n+1)^3.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-09-28.
^ Sloane, N. J. A. (ed.). "Sequence A047966 (a(n) equal to Sum_{ d divides n } q(d), where q(d) is A000009 equal to number of partitions of d into distinct parts.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-09-28.
^ Sloane, N. J. A. (ed.). "Sequence A000695 (Moser-de Bruijn sequence: sums of distinct powers of 4.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-09-28.
^ Sloane, N. J. A. (ed.). "Sequence A053699 (a(n) is n^4 + n^3 + n^2 + n + 1.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-09-28.
^ Sloane, N. J. A. (ed.). "Sequence A000567 (Octagonal numbers: n*(3*n-2). Also called star numbers.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-09-28.
^ Sloane, N. J. A. (ed.). "Sequence A007569 (Number of nodes in regular n-gon with all diagonals drawn.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-09-28.
^ Sloane, N. J. A. (ed.). "Sequence A001045 (Jacobsthal sequence (or Jacobsthal numbers)...)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-09-28.
^ Sloane, N. J. A. (ed.). "Sequence A046746 (Sum of smallest parts of all partitions of n.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-09-28.
^ Sloane, N. J. A. (ed.). "Sequence A003215 (Hex (or centered hexagonal) numbers: 3*n*(n+1)+1 (crystal ball sequence for hexagonal lattice).)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-09-24.
^ Sloane, N. J. A. (ed.). "Sequence A051624 (12-gonal (or dodecagonal) numbers: a(n) equal to n*(5*n-4).)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-09-24.
^ Sloane, N. J. A. (ed.). "Sequence A195316 (Centered 36-gonal numbers.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-09-24.
^ Sloane, N. J. A. (ed.). "Sequence A005936 (Pseudoprimes to base 5.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-09-24.
^ Sloane, N. J. A. (ed.). "Sequence A016105 (Blum integers: numbers of the form p * q where p and q are distinct primes congruent to 3 (mod 4).)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-09-24.
^ Sloane, N. J. A. (ed.). "Sequence A055233 (Composite numbers equal to the sum of the primes from their smallest prime factor to their largest prime factor.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-09-24.
^ Sloane, N. J. A. (ed.). "Sequence A000019 (Number of primitive permutation groups of degree n.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-09-24.
^ Sloane, N. J. A. (ed.). "Sequence A002817 (Doubly triangular numbers: a(n) equal to n*(n+1)*(n^2+n+2)/8.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-09-24.
^ ^a ^b Sloane, N. J. A. (ed.). "Sequence A319274 (Osiris or Digit re-assembly numbers: numbers that are equal to the sum of permutations of subsamples of their own digits.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-09-02.
^ Sloane, N. J. A. (ed.). "Sequence A088825 (Numbers n such that the sum of largest prime factors of numbers from 1 to n is divisible by n.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-09-25.
^ Sloane, N. J. A. (ed.). "Sequence A321719 (Number of non-normal semi-magic squares with sum of entries equal to n.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-08-27.
^ ^a ^b Sloane, N. J. A. (ed.). "Sequence A018784 (Numbers n such that sigma(phi(n)) is n.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-07-16.
^ Sloane, N. J. A. (ed.). "Sequence A017765 (Binomial coefficients C(49,n).)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-09-02.
^ Sloane, N. J. A. (ed.). "Sequence A029600 (Numbers in the (2,3)-Pascal triangle (by row).)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-09-02.
The row is {2, 23, 120, 375, 780, 1134, 1176, 870, 450, 155, 32, 3}.
^ Sloane, N. J. A. (ed.). "Sequence A001263 (Triangle of Narayana numbers T(n,k) is C(n-1,k-1)*C(n,k-1)/k with 1 less than or equal to k less than or equal to n, read by rows. Also called the Catalan triangle.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-09-02.
^ Sloane, N. J. A. (ed.). "Sequence A105278 (Triangle read by rows: T(n,k) equal to binomial(n,k)*(n-1)!/(k-1)!.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-09-02.
^ Sloane, N. J. A. (ed.). "Sequence A127787 (Numbers n such that F(n) divides F(F(n)), where F(n) is a Fibonacci number.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-09-02.
^ Sloane, N. J. A. (ed.). "Sequence A046306 (Numbers that are divisible by exactly 6 primes with multiplicity.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-09-02.
^ Sloane, N. J. A. (ed.). "Sequence A006564 (Icosahedral numbers: a(n) equal to n*(5*n^2 - 5*n + 2)/2.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-09-23.
^ Sloane, N. J. A. (ed.). "Sequence A051177 (Perfectly partitioned numbers: numbers k that divide the number of partitions p(k).)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-09-23.
^ Sloane, N. J. A. (ed.). "Sequence A350028 (Number of Euler tours of the complete graph on n vertices (minus a matching if n is even).)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-07-16.
^ Sloane, N. J. A. (ed.). "Sequence A000014 (Number of series-reduced trees with n nodes.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-09-02.
^ Sloane, N. J. A. (ed.). "Sequence A327070 (Number of non-connected simple labeled graphs covering n vertices.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-08-28.
^ Moree, Pieter (2004). "Convoluted Convolved Fibonacci Numbers" (PDF). Journal of Integer Sequences. 7 (2). Waterloo, Ont., CA: University of Waterloo David R. Cheriton School of Computer Science: 13 (Article 04.2.2). arXiv:math.CO/0311205. Bibcode:2004JIntS...7...22M. MR 2084694. S2CID 14126332. Zbl 1069.11004.
^ ^a ^b Sloane, N. J. A. (ed.). "Sequence A001629 (Self-convolution of Fibonacci numbers.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-07-16.
^ Belbachir, Hacène; Djellal, Toufik; Luque, Jean-Gabriel (2023). "On the self-convolution of generalized Fibonacci numbers". Quaestiones Mathematicae. 46 (5). Oxfordshire, UK: Taylor & Francis: 841–854. arXiv:1703.00323. doi:10.2989/16073606.2022.2043949. S2CID 119150217. Zbl 07707543.{{cite journal}}: CS1 maint: Zbl (link)
^ Sloane, N. J. A. (ed.). "Sequence A004009 (Expansion of Eisenstein series E_4(q) (alternate convention E_2(q)); theta series of E_8 lattice.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-10-05.
^ Sloane, N. J. A. (ed.). "Sequence A013973 (Expansion of Eisenstein series E_6(q) (alternate convention E_3(q)).)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-10-05.
^ Sloane, N. J. A. (ed.). "Sequence A006352 (Coefficients in expansion of Eisenstein series E_2 (also called E_1 or G_2).)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-10-05.
^ Hartshorne, Robin (1977). "Chapter 4: Curves". Algebraic Geometry. Graduate Texts in Mathematics. Vol. 52. Berlin: Springer-Verlag. pp. 316–321. doi:10.1007/978-1-4757-3849-0. ISBN 978-1-4419-2807-8. MR 1567149. OCLC 13348052. Zbl 0367.14001.
^ Berndt, Bruce C.; Chan, Heng Huat (1999). "Ramanujan and the modular j-invariant". Canadian Mathematical Bulletin. 42 (4): 427–440. doi:10.4153/CMB-1999-050-1. MR 1727340. S2CID 1816362.
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^ Barrow, John D. (2002). The Constants of Nature. London: Jonathan Cape. p. 72. doi:10.1142/9789812818201_0001. ISBN 0-224-06135-6. S2CID 125272999.
^ ^a ^b Ronan, Mark (2006). Symmetry and the Monster: One of the Greatest Quests of Mathematics. New York: Oxford University Press. pp. 1–255. doi:10.1007/s00283-008-9007-9. ISBN 978-0-19-280722-9. MR 2215662. OCLC 180766312. Zbl 1113.00002.
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^ Griess, Jr., Robert L. (1982). "The Friendly Giant". Inventiones Mathematicae. 69: 91−96. Bibcode:1982InMat..69....1G. doi:10.1007/BF01389186. hdl:2027.42/46608. MR 0671653. S2CID 123597150. Zbl 0498.20013.
^ Ogg, A. P. (1981). "Modular Functions". In Cooperstein, Bruce; Mason, Geoffrey (eds.). The Santa Cruz Conference on Finite Groups. Proceedings of Symposia in Pure Mathematics. Vol. 37. Providence, RI: American Mathematical Society. pp. 521–532. doi:10.1090/PSPUM/037. ISBN 0-8218-1440-0. MR 0604631. Zbl 0443.00007.
^ Sloane, N. J. A. (ed.). "Sequence A002267 (The 15 supersingular primes: primes dividing order of Monster simple group.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-09-26.
^ ^a ^b Sloane, N. J. A. (ed.). "Sequence A002808 (The composite numbers: numbers n of the form x*y for x > 1 and y > 1.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-09-27.
^ Sloane, N. J. A. (ed.). "Sequence A216071 (Brocard's problem: positive integers m such that m^2 equal to n! + 1 for some n.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-10-10.
^ Sloane, N. J. A. (ed.). "Sequence A085692 (Brocard's problem: squares which can be written as n!+1 for some n.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-10-10.
^ Sloane, N. J. A. (ed.). "Sequence A004394 (Superabundant [or super-abundant] numbers: n such that sigma(n)/n greater than sigma(m)/m for all m less than n, sigma(n) being A000203(n), the sum of the divisors of n.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-10-10.
^ Sloane, N. J. A. (ed.). "Sequence A002182 (Highly composite numbers, definition (1): numbers n where d(n), the number of divisors of n (A000005), increases to a record.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-10-10.
^ Sloane, N. J. A. (ed.). "Sequence A067698 (Positive integers such that sigma(n) greater than or equal to exp(gamma) * n * log(log(n)).)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-10-09.
^ ^a ^b Caveney, Geoffrey; Nicolas, Jean-Louis; Sondow, Jonathan (2011). "Robin's Theorem, Primes, and a New Elementary Reformulation of the Riemann Hypothesis". Integers. 11. Berlin: De Gruyter: 755 (A33). doi:10.1515/INTEG.2011.057. MR 2798609. S2CID 14573608. Zbl 1235.11082.
^ Sloane, N. J. A. (ed.). "Sequence A050381 (Number of series-reduced planted trees with n leaves of 2 colors.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-09-02.
^ ^a ^b Klaise, Janis (2012). Orders in Quadratic Imaginary Fields of small Class Number (PDF) (MMath thesis). University of Warwick Centre for Complexity Science. pp. 1–24. S2CID 126035072.
^ Sloane, N. J. A. (ed.). "Sequence A006450 (Prime-indexed primes: primes with prime subscripts.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-10-18.

^ Weisstein, Eric W. "Almost integer". MathWorld -- A WolframAlpha Resource. Retrieved 2023-10-18.

The list below gives numbers of the form

x \equiv e π \sqrt n

for

n \leq 1000

for which

| nint (x) - x | \leq 10 -3

.

$n$	$\|nint(x) - x \|$
25	−0.00066
37	−0.000022
43	−0.00022
58	−1.8×10⁻⁷
67	−1.3×10⁻⁶
74	−0.00083
148	0.00097
163	−7.5×10⁻¹³
232	−7.8×10⁻⁶
268	0.00029
522	−0.00015
652	1.6×10⁻¹⁰
719	−0.000013

^ ^a ^b Sloane, N. J. A. (ed.). "Sequence A014603 (Discriminants of imaginary quadratic fields with class number 2 (negated).)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-10-18.
^ Sloane, N. J. A. (ed.). "Sequence A032354 (j-invariants for orders of class number 1.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-10-26.
^ Zagier, Don (2008). "Elliptic Modular Forms and Their Applications". In Ranestad, Kristian (ed.). The 1-2-3 of Modular Forms. Universitext (1st ed.). Berlin: Springer-Verlag. pp. 70−71. doi:10.1007/978-3-540-74119-0_1. ISBN 978-3-540-74117-6. MR 2409678. OCLC 173239471. Zbl 1259.11042.
^ Arpin, Sarah; Camacho-Navarro, Catalina; et al. (2021). "Adventures in Supersingularland". Experimental Mathematics. 32 (2). Oxfordshire, UK: Taylor & Francis: 245−256. Bibcode:2019arXiv190907779A. doi:10.1080/10586458.2021.1926009. S2CID 202583238. Zbl 1517.94057.
^ ^a ^b Weisstein, Eric W. (2003). CRC Concise Encyclopedia of Mathematics (2nd ed.). Boca Raton, FL: Chapman & Hall/CRC. pp. 57–59. ISBN 9781584883470. JSTOR 20453513. OCLC 50252094. S2CID 116679721. Zbl 1079.00009.
^ Sloane, N. J. A. (ed.). "Sequence A082897 (Perfect totient numbers.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-10-21.
^ Sloane, N. J. A. (ed.). "Sequence A006203 (Discriminants of imaginary quadratic fields with class number 3 (negated).)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-10-23.
^ Watkins, Mark (2004). "Class numbers of imaginary quadratic fields". Mathematics of Computation. 73 (246). Providence, RI: American Mathematical Society: 907–938. doi:10.1090/S0025-5718-03-01517-5. JSTOR 4099810. MR 2031415. Zbl 1050.11095.
^ Sloane, N. J. A. (ed.). "Sequence A038552 (Largest squarefree number k such that Q(sqrt(-k)) has class number n.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-10-23.
^ Pekonen, Osmo (2018). "Which Integer Is the Most Mysterious?". In Sarhangi, Reza (ed.). Bridges: Mathematics, Art, Music, Architecture, Education, Culture. Proceedings of the Bridges Conference. p. 441. ISSN 1590-5896. OCLC 7788577569. S2CID 133028298.
^ ^a ^b He, Yang-Hui; McKay, John (2015). "Sporadic and Exceptional": 1–49. arXiv:1505.06742. Bibcode:2015arXiv150506742H. S2CID 117030188. {{cite journal}}: Cite journal requires |journal= (help)
"Of particular curiosity is the less well-known fact – in parallel to the above identity – that the constant term of the j-invariant, viz., 744, satisfies $744 = 3 \times 248.$ The number 248 is, of course, the dimension of the adjoint of the largest exceptional algebra [ $𝖊 8$ ]. In fact, that j should encode the presentations of [ $𝖊 8$ ] was settled long before the final proof of the Moonshine conjectures. This relationship between the largest sporadic group and the largest exceptional algebra would connect the McKay Correspondence to Moonshine and thereby weave another beautiful thread into the panoramic tapestry of mathematics."^: p.4

"¹⁶ Incidentally, the reader is also alerted to the curiosity that $σ 1 (240) = 744$ ."^{: p.21 (note)}
^ Sloane, N. J. A. (ed.). "Sequence A007245 (McKay-Thompson series of class 3C for the Monster group.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-10-05.
$j (𝜏) 1/3 = q -1/3 (1 + 248 q + 4124 q 2 + 34752 q 3 + ...)$
^ ^a ^b Gannon, Terry (2006). "Introduction: glimpses of the theory beneath Monstrous Moonshine" (PDF). Moonshine beyond the monster: The bridge connecting algebra, modular forms and physics. Cambridge Monographs on Mathematical Physics. Cambridge, MA: Cambridge University Press. pp. 1–15. ISBN 978-0-521-83531-2. MR 2257727. OCLC 1374925688. Zbl 1146.11026.
"In particular, $4124 = 3875 + 248 + 1$ and $34752 = 30380 + 3875 + 2 \cdot 248 + 1$ , where 248, 3875 and 30380 are all dimensions of irreducible representations of $E 8 (ℂ)$ ."^: p.6
^ Sloane, N. J. A. (ed.). "Sequence A121732 (Dimensions of the irreducible representations of the simple Lie algebra of type E8 over the complex numbers, listed in increasing order.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-10-05.
^ Gaberdiel, Matthias R. (2007). "Constraints on extremal self-dual CFTs". Journal of High Energy Physics (11, 087). Springer: 10–11. Bibcode:2007JHEP...11..087G. doi:10.1088/1126-6708/2007/11/087. S2CID 16635058.
^ Conway, John H.; Sloane, N. J. A. (1988). "Algebraic Constructions for Lattices". Sphere Packings, Lattices and Groups. New York, NY: Springer. doi:10.1007/978-1-4757-2016-7. eISSN 2196-9701. ISBN 978-1-4757-2016-7.
^ ^a ^b ^c Coxeter, H. S. M. (1948). Regular Polytopes (1st ed.). London: Methuen & Co. pp. 1–321. ISBN 9780521201254. MR 0027148. OCLC 472190910.
^ Coxeter, H. S. M. (1998). "Seven Cubes and Ten 24-Cells" (PDF). Discrete & Computational Geometry. 19 (2): 156–157. doi:10.1007/PL00009338. S2CID 206861928. Zbl 0898.52004.
^ Coxeter, H. S. M.; Shephard, G.C. (1992). Portraits of a Family of Complex Polytopes. Vol. 25. Leonardo. pp. 243–244. doi:10.2307/1575843. JSTOR 1575843. S2CID 124245340. Zbl 0803.51023.
^ Baez, John C. (2018). "From the Icosahedron to E₈". London Mathematical Society Newsletter. 476: 18–23. arXiv:1712.06436. MR 3792329. S2CID 119151549. Zbl 1476.51020.
^ Hart, George W. (1998). "Icosahedral Constructions" (PDF). In Sarhangi, Reza (ed.). Bridges: Mathematical Connections in Art, Music, and Science. Proceedings of the Bridges Conference. Winfield, Kansas. p. 196. ISBN 978-0966520101. OCLC 59580549. S2CID 202679388.{{cite book}}: CS1 maint: location missing publisher (link)
^ Wilson, Robert A. (2009). "Octonions and the Leech lattice". Journal of Algebra. 322 (6): 2186–2190. doi:10.1016/j.jalgebra.2009.03.021. MR 2542837.
^ Schellekens, Adrian Norbert (1993). "Meromorphic c = 24 conformal field theories". Communications in Mathematical Physics. 153 (1). Berlin: Springer. Bibcode:1993CMaPh.153..159S. doi:10.1007/BF02099044. MR 1213740. S2CID 250425623. Zbl 0782.17014.
^ Möller, Sven; Scheithauer, Nils R. (2023). "Dimension formulae and generalised deep holes of the Leech lattice vertex operator algebra". Annals of Mathematics. 197 (1). Princeton University & the Institute for Advanced Study: 261–285. Bibcode:2019arXiv191004947M. doi:10.4007/annals.2023.197.1.4. MR 4513145. Zbl 07623024.{{cite journal}}: CS1 maint: Zbl (link)
^ Griess, Jr., Robert L.; Lam, Ching Hung (2011). "A moonshine path from E8 to the Monster" (PDF). Journal of Pure and Applied Algebra. 215 (5): 930–931. doi:10.1016/j.jpaa.2010.07.001. MR 2747229. S2CID 123613651. Zbl 1213.17028.
^ Griess Jr., Robert L.; Lam, Ching Hung (2011). "A moonshine path for 5 A and associated lattices of ranks 8 and 16". Journal of Algebra. 331. Amsterdam: Elsevier: 348. Bibcode:2010arXiv1006.3907G. doi:10.1016/j.jalgebra.2010.11.013. MR 2774662. Zbl 1283.20009.
^ Harada, Masaaki; Lam, Ching Hung; Munemasa, Akihiro (2013). "Residue codes of extremal Type II Z₄-codes and the moonshine vertex operator algebra". Mathematische Zeitschrift. 274 (1). Springer: 691, 698–699. arXiv:1005.1144. Bibcode:2010arXiv1005.1144H. doi:10.1007/s00209-012-1091-z. MR 3054350. S2CID 253730965. Zbl 1283.94112.
^ Pless, V.; Sloane, N. J. A. (1975). "On the classification and enumeration of self-dual codes". Journal of Combinatorial Theory Series A. 18. Amsterdam: Elsevier: 313–335. doi:10.1016/0097-3165(75)90042-4. MR 0376232. S2CID 16303075. Zbl 0305.94011.
^ Van Ekeren, Jethro; Lam, Ching Hung; Möller, Sven; Shimakura, Hiroki (2021). "Schellekens' list and the very strange formula". Advances in Mathematics. 380. Amsterdam: Elsevier: 1–34 (107567). arXiv:2005.12248. doi:10.1016/J.AIM.2021.107567. MR 4200469. S2CID 218870375. Zbl 1492.17027.
^ Betsumiya, Koichi; Lam, Ching Hung; Shimakura, Hiroki (2023). "Automorphism Groups and Uniqueness of Holomorphic Vertex Operator Algebras of Central Charge 24". Communications in Mathematical Physics. 399 (3). Berlin: Springer: 1773–1810. arXiv:2203.15992. Bibcode:2023CMaPh.399.1773B. doi:10.1007/s00220-022-04585-6. S2CID 247793117. Zbl 07681021.{{cite journal}}: CS1 maint: Zbl (link)
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^ Sloane, N. J. A. (ed.). "Sequence A177434 (The magic constants of 6 X 6 magic squares composed of consecutive primes.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-07-16.
^ Sloane, N. J. A. (ed.). "Sequence A177434 (The magic constants of 6 X 6 magic squares composed of consecutive primes.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-09-11.
^ Sloane, N. J. A. (ed.). "Sequence A006003 (a(n) equal to n*(n^2 + 1)/2)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-09-11.
^ Sloane, N. J. A. (ed.). "Sequence A002839 (Number of simple perfect squared rectangles of order n up to symmetry.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-07-16.

[9] 744 is equal to the sum between the 41st and 44th indexed prime numbers, inclusive. Indices 15 and 16 of σ(n) multiply to 240, which is the Euler totient value of 744,^[5] and add to 31 which is the largest prime that is not a totative of 744 (less than).
A sum between 24 and 31 generates the tenth triangular number 55, where the eleventh triangular number is 66.^[6] 121 is the sum between these two triangular numbers, which is equivalent to the square of 11. The prime index of 31 and its permutable prime in decimal (13) form the third pair of twin primes $(11, 13)$ ,^[7] whose sum is 24, with respective prime indices 5 and 6^[8] that add to 11.

[15] 120 is also equal to the sum of the first fifteen integers, or fifteenth triangular number $\textstyle {\sum _{i=1}^{15}i}.$ ^[6] This value is also equal to the sum of all the prime numbers less than 31 that are not factors of 744 except for 5, and including 1. Inclusive of 5, this sum is equal to $125 = 5 3$ , which is the second number after 32 to have an aliquot sum of 31.^[11]
In the Collatz conjecture, 744 and 120 both require fifteen steps to reach 5, before cycling through {16, 8, 4, 2, 1} in five steps.^[12]^[13] Otherwise, they both require twenty steps to reach 1 or nineteen steps to reach 2, which is the middle node in the {1,4,2,1,4...} elementary trajectory for 1 when cycling back to itself.

[21] The radical $186 = 2 \times 3 \times 31$ of 744 has an arithmetic mean of divisors equal to 48,^[10] where the sum between the three distinct prime factors of 744 is $62 = 36$ . 186 is nontotient^[16] and noncototient,^[17] equal to the number of pentahexes when rotations are counted as distinct.^[18]

[27] It is the four hundred and sixth indexed member, where 406 is the twenty-eighth triangular number;^[6] in its base-two representation, the digit positions of zeroes are in $1:1:3$ or $3:1:1$ ratio with the positions of ones, which are in $1:3:1$ ratio. Its ones' complement is 100010111₂, equivalent to $279 = 3 2 \times 31$ in decimal, which represents the sum of GCDs of parts in all partitions of $16 = 4 2$ .^[20] It is also the number of partitions of $62 = 2 \times 31$ (a divisor of 744) as well as 63 into factorial parts (without including 0!),^[21] and the number of integer partitions of 44 whose length is equal to the LCM of all parts^[22] (with 63 the forty-fourth composite number, where 44 is itself the number of derangements of 5).^[23]

[33] On the other hand, every positive integer is at most the sum of two hundred and seventy-nine eighth powers (Waring's problem), preceded by a maximum number of ${1, 4, 9, 19, 37, 73, 143}$ $n$ -powers;^[28] where 279 is the decimal representation of the ones' complement of 744 in binary. In this sequence, the first six members generate a sum equivalent to the seventh member 143, which is the sum of seven consecutive primes starting from 11 through the eleventh prime number: $11 + 13 + 17 + 19 + 23 + 29 + 31$ . It is also the product between the third twin prime pair $(11 \times 13)$ ;^[7] while figuring in $34 + 4 4 + 5 4 + 6 4 = 7 4 - 143$ , which is the first exception to the pattern of polynomials that starts with $32 + 4 2 = 5 2$ and $33 + 4 3 + 5 3 = 6 3$ .

[37] The value of this sigma function represents the fifteenth sum of non-triangular numbers in-between triangular numbers; in this instance it is the sum that lies in-between the fifteenth (120) and sixteenth (136) triangular numbers^[31] (i.e. the sum of $121 + 122 + ... + 135$ ).

[43] The thirtieth triangular number is $465 = 3 \times 5 \times 31$ , equal to the difference between 744 and 279 (equivalently 1011101000₂ and 100010111₂, ones' complement pairs respectively). 30 is the twelfth number $m$ such that $6 m + 1$ and $6 m - 1$ are twin primes (181, 179).^[7]^[34] It is also the binomial $(31,2)$ equal to the number of size-2 subsets of ${0, 1, ..., 31}$ that contain no consecutive integers (in light as a triangular number, trailing its sequence by one index).^[35] In the Padovan sequence, 465 is the twenty-eighth indexed member equal to the number of compositions of 28 into parts congruent to $2 mod 3$ , also the number of compositions of 28 into parts that are odd and greater than or equal to 3, and the number of maximal cliques in a $(28 + 6)$ –path complement graph (specifically, in the $34$ –path).^[36]

[46] The two sets are:
(1, 2, 3, 4, 8, 12, 62, 124, 744)
(6, 24, 31, 186, 93, 248, 372)
960 is also the sum of six consecutive prime numbers $149 + 151 + 157 + 163 + 167 + 173$ , between the 35th and 40th primes (it is the thirty-fifth such number).^[38] The fifteenth and sixteenth triangular numbers generate the sum $120 + 136 = 256 = 2 8$ that is the totient value of 960.^[5]

[9] (1, 2, 3, 4, 8, 12, 62, 124, 744)

[10] (6, 24, 31, 186, 93, 248, 372)

[51] The composite totatives are
{25, 35, 49, 55, 65, 77, 85, 91, 95, 115, 119, 121, 125, 133, 143, 145, 161, 169, 175, 185, 187, 203, 205, 209, 215, 221, 235, 245, 247, 253, 259, 265, 275, 287, 289, 295, 299, 301, 305, 319, 323, 325, 329, 335, 343, 355, 361, 365, 371, 377, 385, 391, 395, 407, 413, 415, 425, 427, 437, 445, 451, 455, 469, 473, 475, 481, 485, 493, 497, 505, 511, 515, 517, 529, 533, 535, 539, 545, 551, 553, 559, 565, 575, 581, 583, 595, 605, 611, 623, 625, 629, 635, 637, 649, 655, 665, 667, 671, 679, 685, 689, 695, 697, 703, 707, 715, 721, 725, 731, 737}.
Its smallest composite totative is $52 = 25$ , that is the only number to have an aliquot sum equal to 6;^[11] where the twenty-fifth prime number 97 is generated from a sum of 48 and 49. This is the first odd prime number that is not a cluster prime^[40] (which precedes the thirty-first prime number 127 in this sequence), where the prime gap for a cluster prime is six or less;^[41] the sixth and seventh primes that are not cluster primes are the forty-eighth and forty-ninth primes 223 and 227. 97 is also the eleventh prime number $p$ of the form $6 m + 1$ for a natural number $m$ .^[42]

[52] 713 is the product of the ninth and eleventh prime numbers $23 \times 31$ , where the twenty-third prime number (83) is the eleventh prime of the form $6 m - 1$ ; it is equal to the difference between 744 and 31, with an aliquot sum of 55^[11] and totient of 660.^[5]

[58] ∗ 589 equal to $19 \times 31$ is the sum of three consecutive primes (193 + 197 + 199).^[43] It is also the ninth centered tetrahedral number, where 5 and 15 are the first two such numbers ( $121 = 11 2$ is the fifth).^[44] It is the fourteenth third spoke of a hexagonal spiral,^[45] and the twentieth quasi-Carmichael number,^[46] with a reduced totient of 90.^[33] In the spt function, 589 is the total number of smallest parts (counted with multiplicity) in all partitions of 15.^[47]

[62] ∗ 527 equal to $17 \times 31$ is the number of partitions of 31 with equal number of even and odd parts.^[48] Its aliquot sum is 49^[11] whose arithmetic mean of divisors is $122 = 144$ , with a sum of its prime factors equal to 48. It is also the maximal number of pieces that can be obtained by cutting an annulus with 31 cuts,^[49] and equivalent to the sum of thirty-one consecutive non-zero integers $2 + 3 + ... + 32$ ; the second-smallest such sum after the third perfect number 496.^[50]

[65] ∗ 403 equal to $13 \times 31$ is the thirty-seventh number to return 0 for the Mertens function,^[51] and the value of the sum-of-divisors of 144.^[52]

[75] ∗ 341 equal to $11 \times 31$ is the sum between seven consecutive primes $(37 + 41 + 43 + 47 + 53 + 59 + 61)$ , of prime indexes 12 through 18. It is the smallest Fermat pseudoprime to binary,^[53] and the sixth centered cube number.^[54] It is the number of partitions of 31 such that every part occurs with the same multiplicity,^[55] and the thirty-first number in the Moser–de Bruijn sequence to be a sum of distinct powers of 4 ( $44 + 4 3 + 4 2 + 4 1 + 4 0$ ).^[56]^[57] 341 is also the eleventh octagonal number,^[58] equal to the number of nodes in a regular eleven-sided undecagon with all diagonals drawn.^[59] $s (341) = 43$ ,^[11] with an arithmetic mean of divisors equal to 96.^[10] It is also the tenth Jacobsthal number,^[60] and the total number of largest parts (or equivalently sum of smallest parts) in all partitions of 16.^[61]

[81] ∗ 217 equal to $7 \times 31$ is the ninth centered hexagonal number, which also includes as previous members {7, 19, 37, 61, 91, 127, 169}^[62] where 127 is the sixth member that is also the thirty-first prime number. 217 is also the sixth non-trivial dodecagonal number^[63] and third non-trivial centered 36-gonal number.^[64] It is the third Fermat pseudoprime to quinary after 124 (a divisor of 744) and 4,^[65] and the fifteenth Blum integer whose distinct prime factors are congruent to $3 mod 4$ .^[66]

[85] ✶ 155 equal to $5 \times 31$ has an aliquot sum of 37^[11] and an arithmetic mean of divisors equal to 48.^[5] It is the third number equal to the sum from its lowest prime factor through its largest, the only smaller numbers with the same property are 10 and 39^[67] which generate a sum of $49 = 7 2$ ; the next such number is 371, 1 less than half of 744. There are one hundred and fifty-five total primitive permutation groups $(G, X)$ of degree $81 = 9 2$ that only preserve partitions on the set $X$ by $G$ that are trivial;^[68] the sum of the number of all permutation groups of lower degrees is 666, which is doubly triangular,^[69] since it represents the sum of the first thirty-six integers, where 36 is itself the eighth triangular number.^[6] The one hundred and fifty-fifth indexed practical number is 744.^[26]

[88] These prime totatives generate a sum of 44,647 while the composite totatives collectively generate a sum of 44,632; these sums have a difference of 15.
On the other hand, differences and sums between the six numbers congruent $\pm1 mod 6$ that are not totatives of 744 are themselves equivalent to divisors of 744, since they are biprimes in proportion with 31 (for example $713 - 589 = 124$ and $713 - 527 = 186$ ), while also generating other relevant equalities such as $713 - 217 = 496$ .
The greatest difference between the largest (713) and smallest (155) of these totatives is 558, the seventh number such that the sum of largest prime factors of numbers from 1 to 558 is divisible by 558, the previous and sixth indexed number is 62,^[71] the tenth-largest divisor of 744, whose sum of divisors is equal to 96;^[52] the fifth number in this sequence is 32 (which divides 96).
The difference between 558 and 62 is also 496, equal to $713 - 155 - 62$ .

[90] The difference between 2238 and 1492 is 746, where 2238 is twice 1119. Specifically, $746 = 2 \times 373 = 1 5 + 2 4 + 3 6 = 2! + 4! + 6!$ is nontotient,^[16] and equal to the number of non-normal orthomagic squares with sum of entries equal to 6.^[72] Its totient value of 372 is half of 744.^[5]

[92] 248 is twice 124 and half 496; the latter which is the third perfect number^[50] — like 6 and 28 (twice 14) — that is also the thirty-first triangular number,^[6] with form $2 p - 1 (2 p - 1)$ and $p = 5$ by the Euclid-Euler theorem. The totient of 248 is 120,^[5] with eight divisors that produce an integer arithmetic mean of 60;^[10] 496 has a totient of 240 like 744.
Where the sum between 744 and the totient $256 = 2 8$ of 960 (the Zumkeller half of 744) is equal to $1000 = 10 3$ , their difference is equivalent with $488 = 240 + 248.$

[99] 1176 is also one of two middle terms in the twelfth row of a $\operatorname {(2,3)} -$ Pascal triangle.^[75] In the triangle of Narayana numbers, 1176 appears as the fortieth and forty-second terms in the eighth row,^[76] which also includes 336 (the totient of 1176)^[5] and 36 (the square of 6). Inside the triangle of Lah numbers of the form $\textstyle {L(n,k)=\left\lfloor {n \atop k}\right\rfloor }$ , 1176 is a member with $n = 8$ and $k = 6$ .^[77] It is a self-Fibonacci number; the fifty-first indexed member where in its case $F_{1176}$ divides $F_{F_{1176}}$ ,^[78] and the forty-first 6-almost prime that is divisible by exactly six primes with multiplicity.^[79]

[102] 456 is the sixth icosahedral number,^[80] that includes 124 as the fourth indexed member in this sequence, which is the eleventh non-unitary divisor of 744 that is equal to the sum of all the prime numbers that are not part of its prime factorization, less than 31 (i.e. $5 + 7 + ... + 29$ ). It is the fourth perfectly partitioned number after 1, 2 and 3.^[81]

[105] Half of 264 is 132, the number of irreducible trees with fifteen vertices.^[83]

[131] 71 is the twentieth prime number and 31 the eleventh. In turn, 20 is the eleventh composite number^[101] that is also the sixth self-convolution of Fibonacci numbers before 38, which is the prime index of 163.
71 is also part of the largest pair of Brown numbers $(71, 7)$ , of only three such pairs; where in its case $72 - 1 = 5040$ .^[102]^[103] Consequently, both 5040 and 5041 can be represented as sums of non-consecutive factorials, following 746, 745, and 744;^[2] where $5040 + 5041 = 10081$ holds an aliquot sum of 611, which is the composite index of 744.^[101]
5040 is the nineteenth superabundant number^[104] that is also the largest factorial that is a highly composite number,^[105] and the largest of twenty-seven numbers $n$ for which the inequality $σ(n) \geq e γ n loglog n$ holds, where $γ$ is the Euler–Mascheroni constant; this inequality is shown to fail for all larger numbers iff the Riemann hypothesis is true (known as Robin's theorem).^[106] 5040 generates a sum-of-divisors $19344 = 13 \times 31 \times 48$ that itself contains four divisors in proportion with 744 (and therefore, divisors also in proportion with 248 as well); which makes it one of only two numbers out of these twenty-seven integers $n$ in Robin's theorem to hold $σ(n)$ such that $744 m$ | $σ(n)$ for any subset of divisors $m$ of $n$ ; the only other such number is 240:^[107]
$19344 \div 26 = 744$
$9672 \div 13 = 744$
$1488 \div 2 = 744$
Where also $19344 \div 78 = 248$ , with 248 and 744 respectively as the 24th and 30th largest divisors (where in between these is $403 = 13 \times 31$ , which is the middle indexed composite number congruent $\pm1 mod 6$ less than 744 that is not part of its composite totatives); 2418, the 35th largest, is the seventh number $n$ after 744 such that $σ (φ (n))$ is $n$ .^[73] Furthermore, in this sequence of integers in Robin's theorem, between 240 and 5040 lie four numbers, where the sum between the first three of these $360 + 720 + 840 = 1920$ is in equivalence with $σ(744).$ The first number to be divisible by all positive non-zero integers less than 11 is the penultimate number in this sequence 2520, where $2520 - 840 - 720 = 960$ represents a Zumkeller half from the set of divisors of 744,^[37] with $σ(720) = 2418$ ^[10] (and while $720 + 24 = 744 = 6! + 24$ ). $5040 = 7! = 10 \times 9 \times 8 \times 7$ is divisible by the first twelve non-zero integers, except for 11.

[26] $19344 \div 26 = 744$

[27] $9672 \div 13 = 744$

[28] $1488 \div 2 = 744$

[147] Otherwise, for almost integers with nearness $|nint(x) - x | \leq 10 -2$ , the smallest such number is $e π \sqrt 6$ with $n$ of 6, approximately equal to 2197.99087.^[116] Whereas 2198 holds eight divisors that produce an arithmetic mean of 474^[10] (and where 474 holds a sum-of-divisors equal to 960,^[52] also the Zumkeller half from the set of divisors of 744),^[37] 2199 is the sixteenth perfect totient number,^[117] with an aliquot sum of 737^[11] equivalent with the largest composite totative of 744. Also, $2197 = 13 3$ .
There are a total of twenty-six almost integers with $|nint(x) - x | \leq 10 -2$ where $x \equiv e π \sqrt n$ and $n \leq 10 3$ , where the largest $n$ holds a value of 986.^[116] The sum between the upper and lower bounds $n$ less than one thousand with this almost integer degree of nearness is equal to $6 + 986 = 992 = 248 \times 4$ .
Also, the difference between the largest class $1$ and $2$ square-free integers is $427 - 163 = 264$ . Regarding square-free integers of class number $h = 3$ , there are a total of sixteen (or twenty-five when including non-maximal orders),^[118] the largest with value of 907 that is the 155th indexed prime number;^[8] of these only 59 produces an almost integer of the form $x \equiv e π \sqrt n$ with $|nint(x) - x | \leq 10 -2$ . In the full list of largest square-free positive integers with class numbers $h \leq 100$ ,^[119]^[120] ten of these maximum values are held by larger integers when non-maximal orders in their respective classes are included, where nine of these are uniquely biprimes divisible by 163 (the only exception is the largest value for square-free integers of class number 16); the largest of these in this bound is $821683 = 71 2 \times 163$ of class number $h = 70$ .^[109]^: 18−20

[177] The magic square is:
${\begin{bmatrix}139&113&151&131&83&127\\223&149&89&47&157&79\\173&103&181&167&59&61\\67&137&53&97&211&179\\101&199&73&109&71&191\\41&43&197&193&163&107\end{bmatrix}}$
This is the second-smallest magic constant for a $6 \times 6$ magic square consisting of thirty-six consecutive prime numbers, the smallest such constant is $484 = 22 2$ ^[145] whose aliquot sum of 447 is the reverse permutation of the digits of 744 in decimal.^[11]
An $11 \times 11$ magic square that is normal has a magic constant of 671,^[146] which is the sixth number to have a sum-of-divisors equal to 744.^[52]

[114] The $j$ –invariant can be computed using Eisenstein series $E 4$ and $E 6$ , such that:
$j(𝜏) = 1728 .mw-parser-output .sfrac{white-space:nowrap}.mw-parser-output .sfrac.tion,.mw-parser-output .sfrac .tion{display:inline-block;vertical-align:-0.5em;font-size:85%;text-align:center}.mw-parser-output .sfrac .num{display:block;line-height:1em;margin:0.0em 0.1em;border-bottom:1px solid}.mw-parser-output .sfrac .den{display:block;line-height:1em;margin:0.1em 0.1em}.mw-parser-output .sr-only{border:0;clip:rect(0,0,0,0);clip-path:polygon(0px 0px,0px 0px,0px 0px);height:1px;margin:-1px;overflow:hidden;padding:0;position:absolute;width:1px}E4(𝜏)3/E4(𝜏)3 − E6(𝜏)2 ,$
where $E 4 (𝜏) = 1 + 240 \sum \infty n =1 (n 3 q n / 1 - q n)$ and $E 6 (𝜏) = 1 - 504 \sum \infty n =1 (n 5 q n / 1 - q n)$ , with $q = exp(2π i 𝜏)$ .
The respective $q$ –expansions of these two Eisenstein series have coefficients whose numerical values are in proportion to 240 and −504, respectively;^[88]^[89] where specifically the sum and difference between the absolute values of these two numbers is $240 + 504 = 744$ and $504 - 240 = 264$ .
Furthermore, when considering the only smaller even (here, non-modular) series $E 2 (𝜏) = 1 - 24 \sum \infty n =1 (nq n / 1 - q n)$ , the sum between the absolute value of its constant multiplicative term (24) and that of $E 4 (𝜏)$ (240) is equal to 264 as well. The 16th coefficient in the expansion of $E 2 (𝜏)$ is −744, as is its 25th coefficient.^[90]
Alternatively, the j–invariant can be computed using a sextic polynomial as: $j (λ) = 2 8 \times (λ 2 - λ + 1) 3 / λ 2 (λ - 2) 2$ where $λ$ represents the λ−modular function, with $256 = 2 8$ .^[91]

[133] The sequence of self-convoluted Fibonacci numbers starts {0, 0, 1, 2, 5, 10, 20, 38, 71, 130, 235, 420, 744, 1308, 2285, 3970, 6865, 11822...}. The sum of the first seven terms (from the zeroth through the sixth term) is equal to 38, which is equivalent to the $a(n) = 7$ member in this sequence. Taking the sum of the three terms that lie between 71^[w] and 744 (i.e. 130, 235, 420) yields 785, whose aliquot sum is 163.^[11] 785 is the 60th number to return $0$ for the Mertens function, which also includes 163, the 13th such number. 785 is also the number of irreducible planted trees (of root vertex having degree one) with six leaves of two colors.^[108]

[136] 163 is the thirty-eighth indexed prime and 67 the nineteenth (with the biprime $38 = 2 \times 19)$ , for values $d$ . In the approximation for the almost integer containing the fourteenth prime number as the seventh Heegner number 43 for $d$ , 960 is equivalent to the sum of either of two sets of divisors of 744 that collectively add to $1920$ , as mentioned in § Totients. Likewise, in the approximation for the almost integer containing the eighth Heegner number 67 for $d$ , 5280 is equal to the sum between 240 and 5040, which are the only two numbers in the set of twenty-seven integers in Robin's theorem for the Riemann hypothesis that have a set of divisors divisible by 744;^[107] 5280 also lies between the 131st pair of twin primes $(5281, 5279)$ ,^[7] respectively the 701st and 700th prime numbers, where 5281 is the 126th super-prime.^[110]

[148] In the list of thirteen integers $n \leq 1000$ that yield almost integers — with nearness $|nint(x) - x | \leq 10 -3$ for values $x \equiv e π \sqrt n$ — the smallest $n$ of these is 25, that is also the smallest composite totative of 744. Less than 10³, the largest such $n$ is $719 = 6! - 1$ , where 719 represents the 128th indexed prime number;^[8] the sum generated by these two $n$ is in equivalence with $719 + 25 = 744.$
The middle indexed value in between these two bounds is 148, the twelfth square-free positive integer $d$ over the negated imaginary quadratic field of class number 2, followed by 163 and 232, the latter of which is the fourteenth square-free positive integer $d$ over the imaginary quadratic field $\sqrt- d$ of class number 2 (only these two numbers 148 and 232 for $d$ in this field of class number 2 yield almost integers with $|nint(x) - x | \leq 10 -3$ , where 163 is the largest over the same field with class number 1).^[112] In this same list, the three distinct non-supersingular primes (37, 43, 67) that only divide orders of pariah groups are also $n$ (less than 148) that yield almost integers of the form $e π \sqrt n$ .
In the almost integer representation for $e π \sqrt 148$ specified, $8 \times 10 3 = j (\sqrt(2i)) = 20 3$ , which is the class $1$ j−invariant of order $d = -8$ ,^[113] that is also a root in $Ψ 2 (X, X) = -(X - 8000) \times (X + 3375) 2 \times (X - 1728)$ , a polynomial for supersingular 2-isogeny graphs with loops (it is also a root in $Ψ 3$ ).^[114]^[115] The remaining two $n$ (53, 74) in the list of these almost integers with nearness $|nint(x) - x | \leq 10 -3$ of $n$ less than 148 generate a sum of $58 + 74 = 132$ equal to the prime index of 743,^[8] the largest prime totative of 744 (and, where $2 \times 37 = 74 = 148 \div 2$ ).^[x]

[161] The 𝔼₈ lattice holds two hundred and forty root vectors that are represented by the vertex arrangement of the $421$ polytope, whose Petrie polygon is a thirty-sided triacontagon,^[129]^[130] where the numerical value of the Carmichael function $λ (n)$ at seven hundred and forty-four is 30,^[33] which is the Coxeter number $h$ of Coxeter group $E 8$ . Also, Coxeter numbers of simple reflexions in $E 6$ and $E 7$ are of orders 12 and 18, together equivalent to the Coxeter number of $E 8$ ;^[128]^: 234 where $E 6$ and $E 7$ are embedded inside $E 8$ . Four-dimensional $H 4$ hexadecachoric symmetry also contains a Coxeter number of 30, where $H 4$ is the higher-dimensional analogue of icosahedral symmetry $I h$ .
This 𝔼₈ lattice structure can be constructed with one hundred and twenty quaternionic unit icosians that form the vertices of the four-dimensional 600-cell, whose symmetries are rooted in three-dimensional icosahedral symmetry $I h$ of order 120;^[131] a value equal to the total number of reflections of Coxeter group $E 8$ ,^[128]^{: 226–232} that is equal to the arithmetic mean of divisors of 744.^[10] In total, a regular icosahedron and dodecahedron contain thirty-one axes of symmetry; six five-fold, ten three-fold, and fifteen two-fold — a count equal to the largest prime factor of 744.^[132] 240, the number of root vectors of $𝔼 8$ , is also the number of integers that are relatively prime with and up to 744,^[5] and the sum-of-divisors of this number is equivalent to $σ(240) = 744$ .^[122]^: p.21

[169] $V E ‍ 3 8$ is isomorphic to the tensor product $V E 8 \otimes V E 8 \otimes V E 8$ ; also, affine structure $D 16$ is different from $D 16 +$ , which is associated with even positive definite unimodular lattice $𝔻 16 +$ .^[136]^[137]
$E ‍ 3 8,1$ and $D 16,1 E 8,1$ are associated with codes $e ‍ 3 8$ and $d 16 e 8$ that are two of only nine in-equivalent doubly even self-dual codes of length 24 and weight 4.^[138]^[139] The largest of these VOAs $V D 24$ is realized in $dim 1128$ , where successively halving its dimensional space leads to a 70½–dimensional space.

[pqr3-1] Sloane, N. J. A. (ed.). "Sequence A189975 (Numbers with prime factorization pqr^3.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-07-16.

[N-CF-2] Sloane, N. J. A. (ed.). "Sequence A060112 (Sums of nonconsecutive factorial numbers.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-07-16.

[3] Sloane, N. J. A. (ed.). "Sequence A034963 (Sums of four consecutive primes.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-07-16.

[4] Sloane, N. J. A. (ed.). "Sequence A083539 (a(n) is sigma(n) * sigma(n+1) as the product of sigma-values for consecutive integers.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-07-16.

[Etot-5] ^ ^a ^b ^c ^d ^e ^f ^g ^h ⁱ ^j Sloane, N. J. A. (ed.). "Sequence A000010 (Euler totient function phi(n): count numbers less than or equal to n and prime to n.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-07-16.

[TriangNum-6] ^ ^a ^b ^c ^d ^e ^f Sloane, N. J. A. (ed.). "Sequence A000217 (Triangular number: a(n) is the binomial(n+1,2) equivalent to n*(n+1)/2 that is 0 + 1 + 2 + ... + n.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-09-02.

[TwinPs-7] Sloane, N. J. A. (ed.). "Sequence A001359 (Lesser of twin primes.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-10-18.

[PNum-8] Sloane, N. J. A. (ed.). "Sequence A000040 (The prime numbers.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-09-06.

[10] Sloane, N. J. A. (ed.). "Sequence A003601 (Numbers n such that the average of the divisors of n is an integer: sigma_0(n) divides sigma_1(n).)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-07-16.

[ArithNum-11] ^ ^a ^b ^c ^d ^e ^f ^g ^h Sloane, N. J. A. (ed.). "Sequence A102187 (Arithmetic means of divisors of arithmetic numbers (arithmetic numbers, A003601, are those for which the average of the divisors is an integer).)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-07-16.

[AliSum-12] ^ ^a ^b ^c ^d ^e ^f ^g ^h ⁱ ^j ^k Sloane, N. J. A. (ed.). "Sequence A001065 (Sum of proper divisors (or aliquot parts) of n: sum of divisors of n that are less than n.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-09-02.

[3x+1-13] Sloane, N. J. A. (ed.). "3x+1 problem". The On-Line Encyclopedia of Integer Sequences. The OEIS Foundation. Retrieved 2023-09-22.

[ColOrb-14] Sloane, N. J. A. (ed.). "Sequence A006577 (Number of halving and tripling steps to reach 1 in '3x+1' problem, or -1 if 1 is never reached.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-09-22.

[16] Sloane, N. J. A. (ed.). "Sequence A000607 (Number of partitions of n into prime parts.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-07-16.

[17] Sloane, N. J. A. (ed.). "Sequence A347586 (Number of partitions of n into at most 4 distinct parts.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-07-16.

[nontot-18] Sloane, N. J. A. (ed.). "Sequence A005277 (Nontotients: even numbers k such that phi(m) equal to k has no solution.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-08-27.

[19] Sloane, N. J. A. (ed.). "Sequence A005278 (Noncototients: numbers k such that x - phi(x) equal to k has no solution.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-09-24.

[20] Sloane, N. J. A. (ed.). "Sequence A001207 (Number of fixed hexagonal polyominoes with n cells.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-09-24.

[22] Sloane, N. J. A. (ed.). "Sequence A052294 (Pernicious numbers: numbers with a prime number of 1's in their binary expansion.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-08-28.

[23] Sloane, N. J. A. (ed.). "Sequence A078392 (Sum of GCD's of parts in all partitions of n.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-10-01.

[24] Sloane, N. J. A. (ed.). "Sequence A064986 (Number of partitions of n into factorial parts (0! not allowed).)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-10-01.

[25] Sloane, N. J. A. (ed.). "Sequence A316433 (Number of integer partitions of n whose length is equal to the LCM of all parts.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-10-01.

[26] Sloane, N. J. A. (ed.). "Sequence A000166 (Subfactorial or rencontres numbers, or derangements: number of permutations of n elements with no fixed points.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-10-01.

[28] Sloane, N. J. A. (ed.). "Sequence A005101 (Abundant numbers (sum of divisors of m exceeds 2m).)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-04-03.
744 is the 181st indexed abundant number.

[29] Sloane, N. J. A. (ed.). "Sequence A005835 (Pseudoperfect (or semiperfect) numbers n: some subset of the proper divisors of n sums to n.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-04-03.
744 is the 183rd semiperfect number.

[PractNum-30] Sloane, N. J. A. (ed.). "Sequence A005153 (Practical numbers)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-04-03.

[31] Sloane, N. J. A. (ed.). "Sequence A345547 (Numbers that are the sum of nine cubes in eight or more ways.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-07-16.

[32] Sloane, N. J. A. (ed.). "Sequence A002804 ((Presumed) solution to Waring's problem: g(n) equal to 2^n + floor((3/2)^n) - 2.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-10-01.

[34] Sloane, N. J. A. (ed.). "Sequence A075308 (Number of n-digit perfect powers.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-07-16.

[35] Sloane, N. J. A. (ed.). "Sequence A034885 (Record values of sigma(n).)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-07-16.

[36] Sloane, N. J. A. (ed.). "Sequence A006002 (a(n) equal to n*(n+1)^2/2 (Sum of the nontriangular numbers between successive triangular numbers. 1, (2), 3, (4, 5), 6, (7, 8, 9), 10, (11, 12, 13, 14), 15, ...))". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-08-27.

[38] Sloane, N. J. A. (ed.). "Sequence A081377 (Numbers n such that the set of prime divisors of phi(n) is equal to the set of prime divisors of sigma(n).)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-07-16.

[redTot-39] Sloane, N. J. A. (ed.). "Sequence A002322 (Reduced totient function psi(n): least k such that x^k is congruent 1 (mod n) for all x prime to n; also known as the Carmichael lambda function (exponent of unit group mod n); also called the universal exponent of n.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-08-27.

[40] Sloane, N. J. A. (ed.). "Sequence A002822 (Numbers m such that 6m-1, 6m+1 are twin primes.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-10-02.

[41] Sloane, N. J. A. (ed.). "Sequence A161680 (a(n) is the binomial(n,2): number of size-2 subsets of {0,1,...,n} that contain no consecutive integers.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-10-02.

[42] Sloane, N. J. A. (ed.). "Sequence A000931 (Padovan sequence (or Padovan numbers)...)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-10-02.

[Zumk-44] Sloane, N. J. A. (ed.). "Sequence A083207 (Zumkeller or integer-perfect numbers: numbers n whose divisors can be partitioned into two disjoint sets with equal sum.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-07-16.

[45] Sloane, N. J. A. (ed.). "Sequence A127333 (Numbers that are the sum of 6 consecutive primes.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-08-27.

[47] Sloane, N. J. A. (ed.). "Sequence A038509 (Composite numbers congruent to +-1 mod 6.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.

[48] Sloane, N. J. A. (ed.). "Sequence A038133 (From a subtractive Goldbach conjecture: odd primes that are not cluster primes.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-09-26.

[49] Blecksmith, Richard; Erdős, Paul; Selfridge, J. L. (1999). "Cluster Primes". The American Mathematical Monthly. 106 (1): 43. doi:10.2307/2589585. JSTOR 2589585. Zbl 0985.11041.

[50] Sloane, N. J. A. (ed.). "Sequence A002476 (Primes of the form 6m + 1.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-09-26.

[53] Sloane, N. J. A. (ed.). "Sequence A034961 (Sums of three consecutive primes.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-09-24.

[54] Sloane, N. J. A. (ed.). "Sequence A005894 (Centered tetrahedral numbers.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-09-24.

[55] Sloane, N. J. A. (ed.). "Sequence A056107 (Third spoke of a hexagonal spiral.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-09-24.

[56] Sloane, N. J. A. (ed.). "Sequence A257750 (Quasi-Carmichael numbers.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-09-24.

[57] Sloane, N. J. A. (ed.). "Sequence A092269 (Spt function: total number of smallest parts (counted with multiplicity) in all partitions of n.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-09-24.

[59] Sloane, N. J. A. (ed.). "Sequence A045931 (Number of partitions of n with equal number of even and odd parts.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-09-24.

[60] Sloane, N. J. A. (ed.). "Sequence A000096 (a(n) equal to n*(n+3)/2.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-09-24.

[PerfN-61] Sloane, N. J. A. (ed.). "Sequence A000396 (Perfect numbers k: k is equal to the sum of the proper divisors of k.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-09-24.

[63] Sloane, N. J. A. (ed.). "Sequence A028442 (Numbers k such that Mertens's function M(k) (A002321) is zero.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-09-24.

[SigmaSoD-64] Sloane, N. J. A. (ed.). "Sequence A000203 (a(n) equal to sigma(n), the sum of the divisors of n. Also called sigma_1(n).)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-09-11.

[66] Sloane, N. J. A. (ed.). "Sequence A001567 (Fermat pseudoprimes to base 2, also called Sarrus numbers or Poulet numbers.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-09-28.

[67] Sloane, N. J. A. (ed.). "Sequence A005898 (Centered cube numbers: n^3 + (n+1)^3.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-09-28.

[68] Sloane, N. J. A. (ed.). "Sequence A047966 (a(n) equal to Sum_{ d divides n } q(d), where q(d) is A000009 equal to number of partitions of d into distinct parts.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-09-28.

[69] Sloane, N. J. A. (ed.). "Sequence A000695 (Moser-de Bruijn sequence: sums of distinct powers of 4.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-09-28.

[70] Sloane, N. J. A. (ed.). "Sequence A053699 (a(n) is n^4 + n^3 + n^2 + n + 1.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-09-28.

[71] Sloane, N. J. A. (ed.). "Sequence A000567 (Octagonal numbers: n*(3*n-2). Also called star numbers.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-09-28.

[72] Sloane, N. J. A. (ed.). "Sequence A007569 (Number of nodes in regular n-gon with all diagonals drawn.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-09-28.

[73] Sloane, N. J. A. (ed.). "Sequence A001045 (Jacobsthal sequence (or Jacobsthal numbers)...)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-09-28.

[74] Sloane, N. J. A. (ed.). "Sequence A046746 (Sum of smallest parts of all partitions of n.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-09-28.

[76] Sloane, N. J. A. (ed.). "Sequence A003215 (Hex (or centered hexagonal) numbers: 3*n*(n+1)+1 (crystal ball sequence for hexagonal lattice).)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-09-24.

[77] Sloane, N. J. A. (ed.). "Sequence A051624 (12-gonal (or dodecagonal) numbers: a(n) equal to n*(5*n-4).)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-09-24.

[78] Sloane, N. J. A. (ed.). "Sequence A195316 (Centered 36-gonal numbers.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-09-24.

[79] Sloane, N. J. A. (ed.). "Sequence A005936 (Pseudoprimes to base 5.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-09-24.

[80] Sloane, N. J. A. (ed.). "Sequence A016105 (Blum integers: numbers of the form p * q where p and q are distinct primes congruent to 3 (mod 4).)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-09-24.

[82] Sloane, N. J. A. (ed.). "Sequence A055233 (Composite numbers equal to the sum of the primes from their smallest prime factor to their largest prime factor.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-09-24.

[83] Sloane, N. J. A. (ed.). "Sequence A000019 (Number of primitive permutation groups of degree n.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-09-24.

[84] Sloane, N. J. A. (ed.). "Sequence A002817 (Doubly triangular numbers: a(n) equal to n*(n+1)*(n^2+n+2)/8.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-09-24.

[Osiris-86] Sloane, N. J. A. (ed.). "Sequence A319274 (Osiris or Digit re-assembly numbers: numbers that are equal to the sum of permutations of subsamples of their own digits.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-09-02.

[87] Sloane, N. J. A. (ed.). "Sequence A088825 (Numbers n such that the sum of largest prime factors of numbers from 1 to n is divisible by n.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-09-25.

[89] Sloane, N. J. A. (ed.). "Sequence A321719 (Number of non-normal semi-magic squares with sum of entries equal to n.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-08-27.

[SigmPhi-91] Sloane, N. J. A. (ed.). "Sequence A018784 (Numbers n such that sigma(phi(n)) is n.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-07-16.

[93] Sloane, N. J. A. (ed.). "Sequence A017765 (Binomial coefficients C(49,n).)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-09-02.

[94] Sloane, N. J. A. (ed.). "Sequence A029600 (Numbers in the (2,3)-Pascal triangle (by row).)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-09-02.
The row is {2, 23, 120, 375, 780, 1134, 1176, 870, 450, 155, 32, 3}.

[95] Sloane, N. J. A. (ed.). "Sequence A001263 (Triangle of Narayana numbers T(n,k) is C(n-1,k-1)*C(n,k-1)/k with 1 less than or equal to k less than or equal to n, read by rows. Also called the Catalan triangle.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-09-02.

[96] Sloane, N. J. A. (ed.). "Sequence A105278 (Triangle read by rows: T(n,k) equal to binomial(n,k)*(n-1)!/(k-1)!.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-09-02.

[97] Sloane, N. J. A. (ed.). "Sequence A127787 (Numbers n such that F(n) divides F(F(n)), where F(n) is a Fibonacci number.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-09-02.

[98] Sloane, N. J. A. (ed.). "Sequence A046306 (Numbers that are divisible by exactly 6 primes with multiplicity.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-09-02.

[100] Sloane, N. J. A. (ed.). "Sequence A006564 (Icosahedral numbers: a(n) equal to n*(5*n^2 - 5*n + 2)/2.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-09-23.

[101] Sloane, N. J. A. (ed.). "Sequence A051177 (Perfectly partitioned numbers: numbers k that divide the number of partitions p(k).)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-09-23.

[103] Sloane, N. J. A. (ed.). "Sequence A350028 (Number of Euler tours of the complete graph on n vertices (minus a matching if n is even).)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-07-16.

[104] Sloane, N. J. A. (ed.). "Sequence A000014 (Number of series-reduced trees with n nodes.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-09-02.

[106] Sloane, N. J. A. (ed.). "Sequence A327070 (Number of non-connected simple labeled graphs covering n vertices.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-08-28.

[107] Moree, Pieter (2004). "Convoluted Convolved Fibonacci Numbers" (PDF). Journal of Integer Sequences. 7 (2). Waterloo, Ont., CA: University of Waterloo David R. Cheriton School of Computer Science: 13 (Article 04.2.2). arXiv:math.CO/0311205. Bibcode:2004JIntS...7...22M. MR 2084694. S2CID 14126332. Zbl 1069.11004.

[ConvFib-108] Sloane, N. J. A. (ed.). "Sequence A001629 (Self-convolution of Fibonacci numbers.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-07-16.

[109] Belbachir, Hacène; Djellal, Toufik; Luque, Jean-Gabriel (2023). "On the self-convolution of generalized Fibonacci numbers". Quaestiones Mathematicae. 46 (5). Oxfordshire, UK: Taylor & Francis: 841–854. arXiv:1703.00323. doi:10.2989/16073606.2022.2043949. S2CID 119150217. Zbl 07707543.{{cite journal}}: CS1 maint: Zbl (link)

[110] Sloane, N. J. A. (ed.). "Sequence A004009 (Expansion of Eisenstein series E_4(q) (alternate convention E_2(q)); theta series of E_8 lattice.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-10-05.

[111] Sloane, N. J. A. (ed.). "Sequence A013973 (Expansion of Eisenstein series E_6(q) (alternate convention E_3(q)).)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-10-05.

[112] Sloane, N. J. A. (ed.). "Sequence A006352 (Coefficients in expansion of Eisenstein series E_2 (also called E_1 or G_2).)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-10-05.

[113] Hartshorne, Robin (1977). "Chapter 4: Curves". Algebraic Geometry. Graduate Texts in Mathematics. Vol. 52. Berlin: Springer-Verlag. pp. 316–321. doi:10.1007/978-1-4757-3849-0. ISBN 978-1-4419-2807-8. MR 1567149. OCLC 13348052. Zbl 0367.14001.

[115] Berndt, Bruce C.; Chan, Heng Huat (1999). "Ramanujan and the modular j-invariant". Canadian Mathematical Bulletin. 42 (4): 427–440. doi:10.4153/CMB-1999-050-1. MR 1727340. S2CID 1816362.

[116] Sloane, N. J. A. (ed.). "Sequence A060295 (Decimal expansion of exp(Pi*sqrt(163)).)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-08-18.

[117] Barrow, John D. (2002). The Constants of Nature. London: Jonathan Cape. p. 72. doi:10.1142/9789812818201_0001. ISBN 0-224-06135-6. S2CID 125272999.

[MarkR-118] Ronan, Mark (2006). Symmetry and the Monster: One of the Greatest Quests of Mathematics. New York: Oxford University Press. pp. 1–255. doi:10.1007/s00283-008-9007-9. ISBN 978-0-19-280722-9. MR 2215662. OCLC 180766312. Zbl 1113.00002.

[119] Ronan, Mark. "163 and the Monster". Mark Ronan. Retrieved 2023-08-17.

[JanChr-120] Jansen, Christoph (2005). "The Minimal Degrees of Faithful Representations of the Sporadic Simple Groups and their Covering Groups". LMS Journal of Computation and Mathematics. 8. London: London Mathematical Society: 122−123. doi:10.1112/S1461157000000930. MR 2153793. S2CID 121362819. Zbl 1089.20006.

[121] Griess, Jr., Robert L. (1982). "The Friendly Giant". Inventiones Mathematicae. 69: 91−96. Bibcode:1982InMat..69....1G. doi:10.1007/BF01389186. hdl:2027.42/46608. MR 0671653. S2CID 123597150. Zbl 0498.20013.

[122] Ogg, A. P. (1981). "Modular Functions". In Cooperstein, Bruce; Mason, Geoffrey (eds.). The Santa Cruz Conference on Finite Groups. Proceedings of Symposia in Pure Mathematics. Vol. 37. Providence, RI: American Mathematical Society. pp. 521–532. doi:10.1090/PSPUM/037. ISBN 0-8218-1440-0. MR 0604631. Zbl 0443.00007.

[123] Sloane, N. J. A. (ed.). "Sequence A002267 (The 15 supersingular primes: primes dividing order of Monster simple group.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-09-26.

[CompNum-124] Sloane, N. J. A. (ed.). "Sequence A002808 (The composite numbers: numbers n of the form x*y for x > 1 and y > 1.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-09-27.

[125] Sloane, N. J. A. (ed.). "Sequence A216071 (Brocard's problem: positive integers m such that m^2 equal to n! + 1 for some n.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-10-10.

[126] Sloane, N. J. A. (ed.). "Sequence A085692 (Brocard's problem: squares which can be written as n!+1 for some n.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-10-10.

[127] Sloane, N. J. A. (ed.). "Sequence A004394 (Superabundant [or super-abundant] numbers: n such that sigma(n)/n greater than sigma(m)/m for all m less than n, sigma(n) being A000203(n), the sum of the divisors of n.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-10-10.

[128] Sloane, N. J. A. (ed.). "Sequence A002182 (Highly composite numbers, definition (1): numbers n where d(n), the number of divisors of n (A000005), increases to a record.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-10-10.

[129] Sloane, N. J. A. (ed.). "Sequence A067698 (Positive integers such that sigma(n) greater than or equal to exp(gamma) * n * log(log(n)).)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-10-09.

[RobThRieH-130] Caveney, Geoffrey; Nicolas, Jean-Louis; Sondow, Jonathan (2011). "Robin's Theorem, Primes, and a New Elementary Reformulation of the Riemann Hypothesis". Integers. 11. Berlin: De Gruyter: 755 (A33). doi:10.1515/INTEG.2011.057. MR 2798609. S2CID 14573608. Zbl 1235.11082.

[132] Sloane, N. J. A. (ed.). "Sequence A050381 (Number of series-reduced planted trees with n leaves of 2 colors.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-09-02.

[KlaiJan-134] Klaise, Janis (2012). Orders in Quadratic Imaginary Fields of small Class Number (PDF) (MMath thesis). University of Warwick Centre for Complexity Science. pp. 1–24. S2CID 126035072.

[135] Sloane, N. J. A. (ed.). "Sequence A006450 (Prime-indexed primes: primes with prime subscripts.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-10-18.

[137] Weisstein, Eric W. "Almost integer". MathWorld -- A WolframAlpha Resource. Retrieved 2023-10-18.
The list below gives numbers of the form $x \equiv e π \sqrt n$ for $n \leq 1000$ for which $| nint (x) - x | \leq 10 -3$ .

$n$ $|nint(x) - x |$

25 −0.00066

37 −0.000022

43 −0.00022

58 −1.8×10⁻⁷

67 −1.3×10⁻⁶

74 −0.00083

148 0.00097

163 −7.5×10⁻¹³

232 −7.8×10⁻⁶

268 0.00029

522 −0.00015

652 1.6×10⁻¹⁰

719 −0.000013

[Cls2-138] Sloane, N. J. A. (ed.). "Sequence A014603 (Discriminants of imaginary quadratic fields with class number 2 (negated).)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-10-18.

[139] Sloane, N. J. A. (ed.). "Sequence A032354 (j-invariants for orders of class number 1.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-10-26.

[140] Zagier, Don (2008). "Elliptic Modular Forms and Their Applications". In Ranestad, Kristian (ed.). The 1-2-3 of Modular Forms. Universitext (1st ed.). Berlin: Springer-Verlag. pp. 70−71. doi:10.1007/978-3-540-74119-0_1. ISBN 978-3-540-74117-6. MR 2409678. OCLC 173239471. Zbl 1259.11042.

[141] Arpin, Sarah; Camacho-Navarro, Catalina; et al. (2021). "Adventures in Supersingularland". Experimental Mathematics. 32 (2). Oxfordshire, UK: Taylor & Francis: 245−256. Bibcode:2019arXiv190907779A. doi:10.1080/10586458.2021.1926009. S2CID 202583238. Zbl 1517.94057.

[CRCEncyc-142] Weisstein, Eric W. (2003). CRC Concise Encyclopedia of Mathematics (2nd ed.). Boca Raton, FL: Chapman & Hall/CRC. pp. 57–59. ISBN 9781584883470. JSTOR 20453513. OCLC 50252094. S2CID 116679721. Zbl 1079.00009.

[143] Sloane, N. J. A. (ed.). "Sequence A082897 (Perfect totient numbers.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-10-21.

[144] Sloane, N. J. A. (ed.). "Sequence A006203 (Discriminants of imaginary quadratic fields with class number 3 (negated).)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-10-23.

[145] Watkins, Mark (2004). "Class numbers of imaginary quadratic fields". Mathematics of Computation. 73 (246). Providence, RI: American Mathematical Society: 907–938. doi:10.1090/S0025-5718-03-01517-5. JSTOR 4099810. MR 2031415. Zbl 1050.11095.

[146] Sloane, N. J. A. (ed.). "Sequence A038552 (Largest squarefree number k such that Q(sqrt(-k)) has class number n.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-10-23.

[149] Pekonen, Osmo (2018). "Which Integer Is the Most Mysterious?". In Sarhangi, Reza (ed.). Bridges: Mathematics, Art, Music, Architecture, Education, Culture. Proceedings of the Bridges Conference. p. 441. ISSN 1590-5896. OCLC 7788577569. S2CID 133028298.

[HeMcKay-150] He, Yang-Hui; McKay, John (2015). "Sporadic and Exceptional": 1–49. arXiv:1505.06742. Bibcode:2015arXiv150506742H. S2CID 117030188. {{cite journal}}: Cite journal requires |journal= (help)
"Of particular curiosity is the less well-known fact – in parallel to the above identity – that the constant term of the j-invariant, viz., 744, satisfies $744 = 3 \times 248.$ The number 248 is, of course, the dimension of the adjoint of the largest exceptional algebra [ $𝖊 8$ ]. In fact, that j should encode the presentations of [ $𝖊 8$ ] was settled long before the final proof of the Moonshine conjectures. This relationship between the largest sporadic group and the largest exceptional algebra would connect the McKay Correspondence to Moonshine and thereby weave another beautiful thread into the panoramic tapestry of mathematics."^: p.4

"¹⁶ Incidentally, the reader is also alerted to the curiosity that $σ 1 (240) = 744$ ."^{: p.21 (note)}

[151] Sloane, N. J. A. (ed.). "Sequence A007245 (McKay-Thompson series of class 3C for the Monster group.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-10-05.
$j (𝜏) 1/3 = q -1/3 (1 + 248 q + 4124 q 2 + 34752 q 3 + ...)$

[TGann-152] Gannon, Terry (2006). "Introduction: glimpses of the theory beneath Monstrous Moonshine" (PDF). Moonshine beyond the monster: The bridge connecting algebra, modular forms and physics. Cambridge Monographs on Mathematical Physics. Cambridge, MA: Cambridge University Press. pp. 1–15. ISBN 978-0-521-83531-2. MR 2257727. OCLC 1374925688. Zbl 1146.11026.
"In particular, $4124 = 3875 + 248 + 1$ and $34752 = 30380 + 3875 + 2 \cdot 248 + 1$ , where 248, 3875 and 30380 are all dimensions of irreducible representations of $E 8 (ℂ)$ ."^: p.6

[153] Sloane, N. J. A. (ed.). "Sequence A121732 (Dimensions of the irreducible representations of the simple Lie algebra of type E8 over the complex numbers, listed in increasing order.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-10-05.

[GebMatt-154] Gaberdiel, Matthias R. (2007). "Constraints on extremal self-dual CFTs". Journal of High Energy Physics (11, 087). Springer: 10–11. Bibcode:2007JHEP...11..087G. doi:10.1088/1126-6708/2007/11/087. S2CID 16635058.

[155] Conway, John H.; Sloane, N. J. A. (1988). "Algebraic Constructions for Lattices". Sphere Packings, Lattices and Groups. New York, NY: Springer. doi:10.1007/978-1-4757-2016-7. eISSN 2196-9701. ISBN 978-1-4757-2016-7.

[HSMRP-156] Coxeter, H. S. M. (1948). Regular Polytopes (1st ed.). London: Methuen & Co. pp. 1–321. ISBN 9780521201254. MR 0027148. OCLC 472190910.

[157] Coxeter, H. S. M. (1998). "Seven Cubes and Ten 24-Cells" (PDF). Discrete & Computational Geometry. 19 (2): 156–157. doi:10.1007/PL00009338. S2CID 206861928. Zbl 0898.52004.

[158] Coxeter, H. S. M.; Shephard, G.C. (1992). Portraits of a Family of Complex Polytopes. Vol. 25. Leonardo. pp. 243–244. doi:10.2307/1575843. JSTOR 1575843. S2CID 124245340. Zbl 0803.51023.

[159] Baez, John C. (2018). "From the Icosahedron to E₈". London Mathematical Society Newsletter. 476: 18–23. arXiv:1712.06436. MR 3792329. S2CID 119151549. Zbl 1476.51020.

[160] Hart, George W. (1998). "Icosahedral Constructions" (PDF). In Sarhangi, Reza (ed.). Bridges: Mathematical Connections in Art, Music, and Science. Proceedings of the Bridges Conference. Winfield, Kansas. p. 196. ISBN 978-0966520101. OCLC 59580549. S2CID 202679388.{{cite book}}: CS1 maint: location missing publisher (link)

[162] Wilson, Robert A. (2009). "Octonions and the Leech lattice". Journal of Algebra. 322 (6): 2186–2190. doi:10.1016/j.jalgebra.2009.03.021. MR 2542837.

[163] Schellekens, Adrian Norbert (1993). "Meromorphic c = 24 conformal field theories". Communications in Mathematical Physics. 153 (1). Berlin: Springer. Bibcode:1993CMaPh.153..159S. doi:10.1007/BF02099044. MR 1213740. S2CID 250425623. Zbl 0782.17014.

[164] Möller, Sven; Scheithauer, Nils R. (2023). "Dimension formulae and generalised deep holes of the Leech lattice vertex operator algebra". Annals of Mathematics. 197 (1). Princeton University & the Institute for Advanced Study: 261–285. Bibcode:2019arXiv191004947M. doi:10.4007/annals.2023.197.1.4. MR 4513145. Zbl 07623024.{{cite journal}}: CS1 maint: Zbl (link)

[165] Griess, Jr., Robert L.; Lam, Ching Hung (2011). "A moonshine path from E8 to the Monster" (PDF). Journal of Pure and Applied Algebra. 215 (5): 930–931. doi:10.1016/j.jpaa.2010.07.001. MR 2747229. S2CID 123613651. Zbl 1213.17028.

[166] Griess Jr., Robert L.; Lam, Ching Hung (2011). "A moonshine path for 5 A and associated lattices of ranks 8 and 16". Journal of Algebra. 331. Amsterdam: Elsevier: 348. Bibcode:2010arXiv1006.3907G. doi:10.1016/j.jalgebra.2010.11.013. MR 2774662. Zbl 1283.20009.

[167] Harada, Masaaki; Lam, Ching Hung; Munemasa, Akihiro (2013). "Residue codes of extremal Type II Z₄-codes and the moonshine vertex operator algebra". Mathematische Zeitschrift. 274 (1). Springer: 691, 698–699. arXiv:1005.1144. Bibcode:2010arXiv1005.1144H. doi:10.1007/s00209-012-1091-z. MR 3054350. S2CID 253730965. Zbl 1283.94112.

[168] Pless, V.; Sloane, N. J. A. (1975). "On the classification and enumeration of self-dual codes". Journal of Combinatorial Theory Series A. 18. Amsterdam: Elsevier: 313–335. doi:10.1016/0097-3165(75)90042-4. MR 0376232. S2CID 16303075. Zbl 0305.94011.

[170] Van Ekeren, Jethro; Lam, Ching Hung; Möller, Sven; Shimakura, Hiroki (2021). "Schellekens' list and the very strange formula". Advances in Mathematics. 380. Amsterdam: Elsevier: 1–34 (107567). arXiv:2005.12248. doi:10.1016/J.AIM.2021.107567. MR 4200469. S2CID 218870375. Zbl 1492.17027.

[171] Betsumiya, Koichi; Lam, Ching Hung; Shimakura, Hiroki (2023). "Automorphism Groups and Uniqueness of Holomorphic Vertex Operator Algebras of Central Charge 24". Communications in Mathematical Physics. 399 (3). Berlin: Springer: 1773–1810. arXiv:2203.15992. Bibcode:2023CMaPh.399.1773B. doi:10.1007/s00220-022-04585-6. S2CID 247793117. Zbl 07681021.{{cite journal}}: CS1 maint: Zbl (link)

[172] Sloane, N. J. A. (ed.). "Sequence A129863 (Sums of three consecutive pentagonal numbers.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-07-16.

[173] Sloane, N. J. A. (ed.). "Sequence A000326 (Pentagonal numbers.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-07-16.

[174] Sloane, N. J. A. (ed.). "Sequence A177434 (The magic constants of 6 X 6 magic squares composed of consecutive primes.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-07-16.

[175] Sloane, N. J. A. (ed.). "Sequence A177434 (The magic constants of 6 X 6 magic squares composed of consecutive primes.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-09-11.

[176] Sloane, N. J. A. (ed.). "Sequence A006003 (a(n) equal to n*(n^2 + 1)/2)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-09-11.

[178] Sloane, N. J. A. (ed.). "Sequence A002839 (Number of simple perfect squared rectangles of order n up to symmetry.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-07-16.

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@@ Line 125: / Line 125: @@
 | style="text-align: center" |−0.000013
 |-
-|}</ref><ref name="Cls2">{{Cite OEIS |A014603 |Discriminants of imaginary quadratic fields with class number 2 (negated). |access-date=2023-10-18 }}</ref>{{efn-lg|1=In the list of ''thirteen'' integers {{math|1=''n'' ≤ 1000 }} that yield almost integers — with [[Floor and ceiling functions#Rounding|nearness]] {{math|1={{pipe}}nint(''x'') − ''x''{{pipe}} ≤ 10<sup>−3</sup> }} for values {{math|1=''x'' ≡ ''e''<sup>''π'''''√'''''n''</sup> }} — the smallest {{math|1=''n'' }} of these is 25, that is also the smallest composite totative of 744. Less than 10<sup>3</sup>, the largest such {{math|1=''n'' }} is {{math|1=719 = 6! − 1 }}, where [[719 (number)|719]] represents the 128th indexed prime number;<ref name="PNum" /> the sum generated by these two {{math|1=''n'' }} is in equivalence with {{math|1=719 + 25 = 744. }} The middle indexed value in between these two bounds is '''[[148 (number)|148]]''', the twelfth square-free positive integer {{math|1=''d'' }} over the negated imaginary quadratic field of class number 2, followed by 163 and [[232 (number)|232]], the latter of which is the fourteenth square-free positive integer {{math|1=''d'' }} over the imaginary quadratic field {{math|1=√−''d'' }} of class number 2 (only these two numbers 148 and 232 for {{math|1=''d'' }} in this field of class number 2 yield almost integers with {{math|1={{pipe}}nint(''x'') − ''x''{{pipe}} ≤ 10<sup>−3</sup> }}, where 163 is the largest over the same field with class number 1).<ref name="Cls2" /> In this same list, the three distinct ''non-supersingular primes'' (37, 43, 67) that only divide orders of ''[[pariah group]]s'' are also {{math|1=''n'' }} (less than 148) that yield almost integers of the form {{math|1=''e''<sup>''π'''''√'''''n''</sup> }}; the remaining two {{math|1=''n'' }} in between generate a sum of {{math|1=58 + 74 = 132 }} equal to the prime index of 743,<ref name="PNum" /> the largest prime totative of 744 (and, where {{math|1=2 × 37 = 74 = 148 ÷ 2 }}).{{efn-la|1=Otherwise, for almost integers with nearness {{math|1={{pipe}}nint(''x'') − ''x''{{pipe}} ≤ 10<sup>−2</sup> }}, the smallest such number is {{math|1=''e''<sup>''π'''''√'''''6''</sup> }} with {{math|1=''n'' }} of 6, approximately equal to 2197.99087.<ref name="CRCEncyc">{{Cite book |last=Weisstein |first=Eric W. |author-link=Eric W. Weisstein |title=[[CRC Concise Encyclopedia of Mathematics]] |publisher=[[Chapman & Hall|Chapman & Hall/CRC]] |location=Boca Raton, FL |edition=2nd |pages=57–59|year=2003 |isbn=9781584883470 |oclc=50252094 |jstor=20453513 |s2cid=116679721 |zbl=1079.00009 }}</ref> Whereas 2198 holds eight divisors that produce an arithmetic mean of 474<ref name="ArithNum" /> (and where 474 holds a sum-of-divisors equal to 960,<ref name="SigmaSoD" /> also the ''Zumkeller half'' from the set of divisors of 744),<ref name="Zumk" /> 2199 is the sixteenth [[perfect totient number]],<ref>{{Cite OEIS |A082897 |Perfect totient numbers. |access-date=2023-10-21  }}</ref> with an aliquot sum of [[737 (number)|737]]<ref name="AliSum" /> equivalent with the largest composite totative of 744. Also, {{math|1=2197 = 13<sup>3</sup> }}.<br /> There are a total of ''twenty-six'' almost integers with {{math|1={{pipe}}nint(''x'') − ''x''{{pipe}} ≤ 10<sup>−2</sup> }} where {{math|1=''x'' ≡ ''e''<sup>''π'''''√'''''n''</sup> }} and {{math|1=''n'' ≤ 10<sup>3</sup>}}, where the largest {{math|1=''n'' }} holds a value of [[986 (number)|986]].<ref name="CRCEncyc" /> The sum between the upper and lower bounds {{math|1=''n'' }} less than one thousand with this ''almost integer'' degree of nearness is equal to {{math|1=6 + 986 = '''[[992 (number)|992]]''' = 248 × 4 }}.<br />Also, the difference between the largest class {{math|1=1 }} and {{math|1=2 }} square-free integers is {{math|1=427 − 163 = 264 }}. Regarding square-free integers of class number {{math|1='''''h''''' = 3 }}, there are a total of sixteen (or twenty-five when including non-maximal orders),<ref>{{Cite OEIS |A006203 |Discriminants of imaginary quadratic fields with class number 3 (negated). |access-date=2023-10-23 }}</ref> the largest with value of [[907 (number)|907]] that is the 155th indexed prime number;<ref name="PNum" /> of these only [[59 (number)|59]] produces an almost integer of the form {{math|1=''x'' ≡ ''e''<sup>''π'''''√'''''n''</sup> }} with {{math|1={{pipe}}nint(''x'') − ''x''{{pipe}} ≤ 10<sup>−2</sup> }}. In the full list of largest square-free positive integers with class numbers {{math|1=''h'' ≤ 100 }},<ref>{{Cite journal |last=Watkins |first=Mark |title=Class numbers of imaginary quadratic fields |url=https://www.ams.org/journals/mcom/2004-73-246/S0025-5718-03-01517-5/ |journal=[[Mathematics of Computation]] |publisher=[[American Mathematical Society]] |location=Providence, RI |volume=73 |number=246 |year=2004 |pages=907-938 |doi=10.1090/S0025-5718-03-01517-5 |jstor=4099810 |mr=2031415 |zbl=1050.11095 }}</ref><ref>{{Cite OEIS |A038552 |Largest squarefree number k such that Q(sqrt(-k)) has class number n. |access-date=2023-10-23 }}</ref> ten of these maximum values are held by larger integers when ''non-maximal'' orders in their respective classes are included, where ''nine'' of these are uniquely biprimes divisible by 163 (the only exception is the largest value for square-free integers of class number 16); the largest of these in this bound is {{math|1='''821683''' = 71<sup>2</sup> × 163 }} of class number {{math|1=''h'' = 70 }}.<ref name="KlaiJan" />{{rp|18−20 }} }} }}
+|}</ref><ref name="Cls2">{{Cite OEIS |A014603 |Discriminants of imaginary quadratic fields with class number 2 (negated). |access-date=2023-10-18 }}</ref>{{efn-lg|1=In the list of ''thirteen'' integers {{math|1=''n'' ≤ 1000 }} that yield almost integers — with [[Floor and ceiling functions#Rounding|nearness]] {{math|1={{pipe}}nint(''x'') − ''x''{{pipe}} ≤ 10<sup>−3</sup> }} for values {{math|1=''x'' ≡ ''e''<sup>''π'''''√'''''n''</sup> }} — the smallest {{math|1=''n'' }} of these is 25, that is also the smallest composite totative of 744. Less than 10<sup>3</sup>, the largest such {{math|1=''n'' }} is {{math|1=719 = 6! − 1 }}, where [[719 (number)|719]] represents the 128th indexed prime number;<ref name="PNum" /> the sum generated by these two {{math|1=''n'' }} is in equivalence with {{math|1=719 + 25 = 744. }}<br /> The middle indexed value in between these two bounds is '''[[148 (number)|148]]''', the twelfth square-free positive integer {{math|1=''d'' }} over the negated imaginary quadratic field of class number 2, followed by 163 and [[232 (number)|232]], the latter of which is the fourteenth square-free positive integer {{math|1=''d'' }} over the imaginary quadratic field {{math|1=√−''d'' }} of class number 2 (only these two numbers 148 and 232 for {{math|1=''d'' }} in this field of class number 2 yield almost integers with {{math|1={{pipe}}nint(''x'') − ''x''{{pipe}} ≤ 10<sup>−3</sup> }}, where 163 is the largest over the same field with class number 1).<ref name="Cls2" /> In this same list, the three distinct ''non-supersingular primes'' (37, 43, 67) that only divide orders of ''[[pariah group]]s'' are also {{math|1=''n'' }} (less than 148) that yield almost integers of the form {{math|1=''e''<sup>''π'''''√'''''n''</sup> }}.<br />In the almost integer representation for {{math|1=''e''<sup>''π'''''√'''148</sup> }} specified, {{math|1=8 × 10<sup>3</sup> = '''''j'''''(√(2i)) = 20<sup>3</sup> }}, which is the class {{math|1=1 }} '''''j'''''−invariant of order {{math|1=''d'' = −8 }},<ref>{{Cite OEIS |A032354 |j-invariants for orders of class number 1. |access-date=2023-10-26 }}</ref> that is also a root in {{math|1='''''Ψ'''''<sub>2</sub>(''X'', ''X'') = −(''X'' − 8000) × (''X'' + 3375)<sup>2</sup> × (''X'' − 1728) }}, a polynomial for ''[[Supersingular isogeny graph|supersingular 2-isogeny graphs]]'' with [[Loop (graph theory)|loops]] (it is also a root in {{math|1=''Ψ''<sub>3</sub> }}).<ref>{{Cite book |author-last=Zagier |author-first=Don |author-link=Don Zagier  |editor-last=Ranestad |editor-first=Kristian |chapter=Elliptic Modular Forms and Their Applications |title=The 1-2-3 of Modular Forms |chapter-url=https://link.springer.com/chapter/10.1007/978-3-540-74119-0_1 |edition=1st |series=Universitext |publisher=[[Springer Science+Business Media|Springer-Verlag]] |location=Berlin |year=2008 |pages=70−71 |doi=10.1007/978-3-540-74119-0_1 |isbn=978-3-540-74117-6 |oclc=173239471 |mr=2409678 |zbl=1259.11042 }}</ref><ref>{{Cite journal |last1=Arpin |first1=Sarah |last2=Camacho-Navarro |first2=Catalina  |last3=Lauter |first3=Kristin  |last4=Lim |first4=Joelle |last5=Nelson |first5=Kristina |last6=Scholl |first6=Travis |last7=Sotáková |first7=Jana |display-authors=2 |title=Adventures in Supersingularland |url=https://www.tandfonline.com/doi/full/10.1080/10586458.2021.1926009 |journal=Experimental Mathematics |publisher=[[Taylor & Francis]] |volume=32 |issue=2 |location=Oxfordshire, UK |year=2021 |pages=245−256 |doi=10.1080/10586458.2021.1926009|doi-access=free |bibcode=2019arXiv190907779A |zbl=1517.94057 |s2cid=202583238 }}</ref> The remaining two {{math|1=''n'' }} (53, 74) in the list of these almost integers with nearness {{math|1={{pipe}}nint(''x'') − ''x''{{pipe}} ≤ 10<sup>−3</sup> }} of {{math|1=''n'' }} less than 148 generate a sum of {{math|1=58 + 74 = 132 }} equal to the prime index of 743,<ref name="PNum" /> the largest prime totative of 744 (and, where {{math|1=2 × 37 = 74 = 148 ÷ 2 }}).{{efn-la|1=Otherwise, for almost integers with nearness {{math|1={{pipe}}nint(''x'') − ''x''{{pipe}} ≤ 10<sup>−2</sup> }}, the smallest such number is {{math|1=''e''<sup>''π'''''√'''''6''</sup> }} with {{math|1=''n'' }} of 6, approximately equal to 2197.99087.<ref name="CRCEncyc">{{Cite book |last=Weisstein |first=Eric W. |author-link=Eric W. Weisstein |title=[[CRC Concise Encyclopedia of Mathematics]] |publisher=[[Chapman & Hall|Chapman & Hall/CRC]] |location=Boca Raton, FL |edition=2nd |pages=57–59|year=2003 |isbn=9781584883470 |oclc=50252094 |jstor=20453513 |s2cid=116679721 |zbl=1079.00009 }}</ref> Whereas 2198 holds eight divisors that produce an arithmetic mean of 474<ref name="ArithNum" /> (and where 474 holds a sum-of-divisors equal to 960,<ref name="SigmaSoD" /> also the ''Zumkeller half'' from the set of divisors of 744),<ref name="Zumk" /> 2199 is the sixteenth [[perfect totient number]],<ref>{{Cite OEIS |A082897 |Perfect totient numbers. |access-date=2023-10-21  }}</ref> with an aliquot sum of [[737 (number)|737]]<ref name="AliSum" /> equivalent with the largest composite totative of 744. Also, {{math|1=2197 = 13<sup>3</sup> }}.<br /> There are a total of ''twenty-six'' almost integers with {{math|1={{pipe}}nint(''x'') − ''x''{{pipe}} ≤ 10<sup>−2</sup> }} where {{math|1=''x'' ≡ ''e''<sup>''π'''''√'''''n''</sup> }} and {{math|1=''n'' ≤ 10<sup>3</sup>}}, where the largest {{math|1=''n'' }} holds a value of [[986 (number)|986]].<ref name="CRCEncyc" /> The sum between the upper and lower bounds {{math|1=''n'' }} less than one thousand with this ''almost integer'' degree of nearness is equal to {{math|1=6 + 986 = '''[[992 (number)|992]]''' = 248 × 4 }}.<br />Also, the difference between the largest class {{math|1=1 }} and {{math|1=2 }} square-free integers is {{math|1=427 − 163 = 264 }}. Regarding square-free integers of class number {{math|1='''''h''''' = 3 }}, there are a total of sixteen (or twenty-five when including non-maximal orders),<ref>{{Cite OEIS |A006203 |Discriminants of imaginary quadratic fields with class number 3 (negated). |access-date=2023-10-23 }}</ref> the largest with value of [[907 (number)|907]] that is the 155th indexed prime number;<ref name="PNum" /> of these only [[59 (number)|59]] produces an almost integer of the form {{math|1=''x'' ≡ ''e''<sup>''π'''''√'''''n''</sup> }} with {{math|1={{pipe}}nint(''x'') − ''x''{{pipe}} ≤ 10<sup>−2</sup> }}. In the full list of largest square-free positive integers with class numbers {{math|1=''h'' ≤ 100 }},<ref>{{Cite journal |last=Watkins |first=Mark |title=Class numbers of imaginary quadratic fields |url=https://www.ams.org/journals/mcom/2004-73-246/S0025-5718-03-01517-5/ |journal=[[Mathematics of Computation]] |publisher=[[American Mathematical Society]] |location=Providence, RI |volume=73 |number=246 |year=2004 |pages=907-938 |doi=10.1090/S0025-5718-03-01517-5 |jstor=4099810 |mr=2031415 |zbl=1050.11095 }}</ref><ref>{{Cite OEIS |A038552 |Largest squarefree number k such that Q(sqrt(-k)) has class number n. |access-date=2023-10-23 }}</ref> ten of these maximum values are held by larger integers when ''non-maximal'' orders in their respective classes are included, where ''nine'' of these are uniquely biprimes divisible by 163 (the only exception is the largest value for square-free integers of class number 16); the largest of these in this bound is {{math|1='''821683''' = 71<sup>2</sup> × 163 }} of class number {{math|1=''h'' = 70 }}.<ref name="KlaiJan" />{{rp|18−20 }} }} }}
 === E<sub>8</sub> and the Leech lattice ===

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