# List of OEIS sequences

This article provides a list of integer sequences in the On-Line Encyclopedia of Integer Sequences that have their own English Wikipedia entries.

OEIS link Name First elements Short description
A000002 Kolakoski sequence {1, 2, 2, 1, 1, 2, 1, 2, 2, 1, …} The nth term describes the length of the nth run
A000010 Euler's totient function φ(n) {1, 1, 2, 2, 4, 2, 6, 4, 6, 4, …} φ(n) is the number of positive integers not greater than n that are prime to n.
A000027 Natural numbers {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, …} The natural numbers (positive integers) n ∈ ℕ.
A000032 Lucas numbers L(n) {2, 1, 3, 4, 7, 11, 18, 29, 47, 76, …} L(n) = L(n − 1) + L(n − 2) for n ≥ 2, with L(0) = 2 and L(1) = 1.
A000040 Prime numbers pn {2, 3, 5, 7, 11, 13, 17, 19, 23, 29, …} The prime numbers pn, with n ≥ 1.
A000041 Partition numbers
Pn
{1, 1, 2, 3, 5, 7, 11, 15, 22, 30, 42, ...} The partition numbers, number of additive breakdowns of n.
A000045 Fibonacci numbers F(n) {0, 1, 1, 2, 3, 5, 8, 13, 21, 34, …} F(n) = F(n − 1) + F(n − 2) for n ≥ 2, with F(0) = 0 and F(1) = 1.
A000058 Sylvester's sequence {2, 3, 7, 43, 1807, 3263443, 10650056950807, 113423713055421844361000443, …} a(n + 1) = a(n)⋅a(n − 1)⋅ ⋯ ⋅a(0) + 1 = a(n)2a(n) + 1 for n ≥ 1, with a(0) = 2.
A000073 Tribonacci numbers {0, 1, 1, 2, 4, 7, 13, 24, 44, 81, …} T(n) = T(n − 1) + T(n − 2) + T(n − 3) for n ≥ 3, with T(0) = 0 and T(1) = T(2) = 1.
A000108 Catalan numbers Cn {1, 1, 2, 5, 14, 42, 132, 429, 1430, 4862, …} ${\displaystyle C_{n}={\frac {1}{n+1}}{2n \choose n}={\frac {(2n)!}{(n+1)!\,n!}}=\prod \limits _{k=2}^{n}{\frac {n+k}{k}},\quad n\geq 0.}$
A000110 Bell numbers Bn {1, 1, 2, 5, 15, 52, 203, 877, 4140, 21147, …} Bn is the number of partitions of a set with n elements.
A000111 Euler zigzag numbers En {1, 1, 1, 2, 5, 16, 61, 272, 1385, 7936, …} En is the number of linear extensions of the "zig-zag" poset.
A000124 Lazy caterer's sequence {1, 2, 4, 7, 11, 16, 22, 29, 37, 46, …} The maximal number of pieces formed when slicing a pancake with n cuts.
A000129 Pell numbers Pn {0, 1, 2, 5, 12, 29, 70, 169, 408, 985, …} a(n) = 2a(n − 1) + a(n − 2) for n ≥ 2, with a(0) = 0, a(1) = 1.
A000142 Factorials n! {1, 1, 2, 6, 24, 120, 720, 5040, 40320, 362880, …} n! := 1⋅2⋅3⋅4⋅ ⋯ ⋅n for n ≥ 1, with 0! = 1 (empty product).
A000203 Divisor function σ(n) {1, 3, 4, 7, 6, 12, 8, 15, 13, 18, 12, 28, …} σ(n) := σ1(n) is the sum of divisors of a positive integer n.
A000215 Fermat numbers Fn {3, 5, 17, 257, 65537, 4294967297, 18446744073709551617, 340282366920938463463374607431768211457, …} Fn = 22n + 1 for n ≥ 0.
A000217 Triangular numbers t(n) {0, 1, 3, 6, 10, 15, 21, 28, 36, 45, …} t(n) = C(n + 1, 2) = n (n + 1)/2 = 1 + 2 + ⋯ + n for n ≥ 1, with t(0) = 0 (empty sum).
A000290 Square numbers n2 {0, 1, 4, 9, 16, 25, 36, 49, 64, 81, ...} n2 = n × n
A000292 Tetrahedral numbers T(n) {0, 1, 4, 10, 20, 35, 56, 84, 120, 165, …} T(n) is the sum of the first n triangular numbers, with T(0) = 0 (empty sum).
A000330 Square pyramidal numbers {0, 1, 5, 14, 30, 55, 91, 140, 204, 285, …} n (n + 1)(2n + 1)/6 : The number of stacked spheres in a pyramid with a square base.
A000396 Perfect numbers {6, 28, 496, 8128, 33550336, 8589869056, 137438691328, 2305843008139952128, …} n is equal to the sum s(n) = σ(n) − n of the proper divisors of n.
A000578 Cube numbers n3 {0, 1, 8, 27, 64, 125, 216, 343, 512, 729, ...} n3 = n × n × n
A000668 Mersenne primes {3, 7, 31, 127, 8191, 131071, 524287, 2147483647, 2305843009213693951, 618970019642690137449562111, …} 2p − 1 is prime, where p is a prime.
A000793 Landau's function {1, 1, 2, 3, 4, 6, 6, 12, 15, 20, …} The largest order of permutation of n elements.
A000796 Decimal expansion of π {3, 1, 4, 1, 5, 9, 2, 6, 5, 3, …} Ratio of a circle's circumference to its diameter.
A000930 Narayana's cows {1, 1, 1, 2, 3, 4, 6, 9, 13, 19, …} The number of cows each year if each cow has one cow a year beginning its fourth year.
A000931 Padovan sequence {1, 1, 1, 2, 2, 3, 4, 5, 7, 9, …} P(n) = P(n − 2) + P(n − 3) for n ≥ 3, with P(0) = P(1) = P(2) = 1.
A000945 Euclid–Mullin sequence {2, 3, 7, 43, 13, 53, 5, 6221671, 38709183810571, 139, …} a(1) = 2; a(n + 1) is smallest prime factor of a(1) a(2) ⋯ a(n) + 1.
A000959 Lucky numbers {1, 3, 7, 9, 13, 15, 21, 25, 31, 33, …} A natural number in a set that is filtered by a sieve.
A001006 Motzkin numbers {1, 1, 2, 4, 9, 21, 51, 127, 323, 835, …} The number of ways of drawing any number of nonintersecting chords joining n (labeled) points on a circle.
A001045 Jacobsthal numbers {0, 1, 1, 3, 5, 11, 21, 43, 85, 171, 341, …} a(n) = a(n − 1) + 2a(n − 2) for n ≥ 2, with a(0) = 0, a(1) = 1.
A001065 Sum of proper divisors s(n) {0, 1, 1, 3, 1, 6, 1, 7, 4, 8, …} s(n) = σ(n) − n is the sum of the proper divisors of the positive integer n.
A001113 Decimal expansion of e {2, 7, 1, 8, 2, 8, 1, 8, 2, 8, …} Euler's number in base 10.
A001190 Wedderburn–Etherington numbers {0, 1, 1, 1, 2, 3, 6, 11, 23, 46, …} The number of binary rooted trees (every node has out-degree 0 or 2) with n endpoints (and 2n − 1 nodes in all).
A001220 Wieferich primes {1093, 3511} Primes ${\displaystyle p}$ satisfying 2p-1 ≡ 1 (mod p2).
A001263 Narayana numbers {1, 1, 1, 1, 3, 1, 1, 6, 6, 1, 1, 10, 20, 10, 1, …} ${\displaystyle {1 \over k}{n-1 \choose k-1}{n \choose k-1}}$ read by rows.
A001358 Semiprimes {4, 6, 9, 10, 14, 15, 21, 22, 25, 26, …} Products of two primes, not necessarily distinct.
A001462 Golomb sequence {1, 2, 2, 3, 3, 4, 4, 4, 5, 5, …} a(n) is the number of times n occurs, starting with a(1) = 1.
A001608 Perrin numbers Pn {3, 0, 2, 3, 2, 5, 5, 7, 10, 12, …} P(n) = P(n−2) + P(n−3) for n ≥ 3, with P(0) = 3, P(1) = 0, P(2) = 2.
A001620 Euler–Mascheroni constant γ {5, 7, 7, 2, 1, 5, 6, 6, 4, 9, …} ${\displaystyle \gamma =\lim _{n\rightarrow \infty }\left(\sum _{k=1}^{n}{\frac {1}{k}}-\ln(n)\right)=\lim _{b\rightarrow \infty }\int _{1}^{b}\left({1 \over \lfloor x\rfloor }-{1 \over x}\right)\,dx.}$
A001622 Decimal expansion of the golden ratio φ {1, 6, 1, 8, 0, 3, 3, 9, 8, 8, …} φ = 1 + 5/2 = 1.6180339887... in base 10.
A002064 Cullen numbers Cn {1, 3, 9, 25, 65, 161, 385, 897, 2049, 4609, 10241, 22529, 49153, 106497, …} Cn = n⋅2n + 1, with n ≥ 0.
A002110 Primorials pn# {1, 2, 6, 30, 210, 2310, 30030, 510510, 9699690, 223092870, …} pn#, the product of the first n primes.
A002113 Palindromic numbers {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, …} A number that remains the same when its digits are reversed.
A002182 Highly composite numbers {1, 2, 4, 6, 12, 24, 36, 48, 60, 120, …} A positive integer with more divisors than any smaller positive integer.
A002193 Decimal expansion of 2 {1, 4, 1, 4, 2, 1, 3, 5, 6, 2, …} Square root of 2.
A002201 Superior highly composite numbers {2, 6, 12, 60, 120, 360, 2520, 5040, 55440, 720720, …} A positive integer n for which there is an e > 0 such that d(n)/ned(k)/ke for all k > 1.
A002378 Pronic numbers {0, 2, 6, 12, 20, 30, 42, 56, 72, 90, …} 2t(n) = n (n + 1), with n ≥ 0.
A002559 Markov numbers {1, 2, 5, 13, 29, 34, 89, 169, 194, …} Positive integer solutions of x2 + y2 + z2 = 3xyz.
A002808 Composite numbers {4, 6, 8, 9, 10, 12, 14, 15, 16, 18, …} The numbers n of the form xy for x > 1 and y > 1.
A002858 Ulam number {1, 2, 3, 4, 6, 8, 11, 13, 16, 18, …} a(1) = 1; a(2) = 2; for n > 2, a(n) is least number > a(n − 1) which is a unique sum of two distinct earlier terms; semiperfect.
A002997 Carmichael numbers {561, 1105, 1729, 2465, 2821, 6601, 8911, 10585, 15841, 29341, …} Composite numbers n such that an − 1 ≡ 1 (mod n) if a is prime to n.
A003261 Woodall numbers {1, 7, 23, 63, 159, 383, 895, 2047, 4607, …} n⋅2n − 1, with n ≥ 1.
A003459 Permutable primes {2, 3, 5, 7, 11, 13, 17, 31, 37, 71, …} The numbers for which every permutation of digits is a prime.
A005044 Alcuin's sequence {0, 0, 0, 1, 0, 1, 1, 2, 1, 3, 2, 4, 3, 5, 4, 7, 5, 8, 7, 10, 8, 12, 10, 14, …} Number of triangles with integer sides and perimeter n.
A005100 Deficient numbers {1, 2, 3, 4, 5, 7, 8, 9, 10, 11, …} Positive integers n such that σ(n) < 2n.
A005101 Abundant numbers {12, 18, 20, 24, 30, 36, 40, 42, 48, 54, …} Positive integers n such that σ(n) > 2n.
A005114 Untouchable numbers {2, 5, 52, 88, 96, 120, 124, 146, 162, 188, …} Cannot be expressed as the sum of all the proper divisors of any positive integer.
A005150 Look-and-say sequence {1, 11, 21, 1211, 111221, 312211, 13112221, 1113213211, 31131211131221, 13211311123113112211, …} A = 'frequency' followed by 'digit'-indication.
A005224 Aronson's sequence {1, 4, 11, 16, 24, 29, 33, 35, 39, 45, …} "t" is the first, fourth, eleventh, ... letter in this sentence, not counting spaces or commas.
A005235 Fortunate numbers {3, 5, 7, 13, 23, 17, 19, 23, 37, 61, …} The smallest integer m > 1 such that pn# + m is a prime number, where the primorial pn# is the product of the first n prime numbers.
A005349 Harshad numbers in base 10 {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, …} A Harshad number in base 10 is an integer that is divisible by the sum of its digits (when written in base 10).
A005384 Sophie Germain primes {2, 3, 5, 11, 23, 29, 41, 53, 83, 89, …} A prime number p such that 2p + 1 is also prime.
A005835 Semiperfect numbers {6, 12, 18, 20, 24, 28, 30, 36, 40, 42, …} A natural number n that is equal to the sum of all or some of its proper divisors.
A006037 Weird numbers {70, 836, 4030, 5830, 7192, 7912, 9272, 10430, 10570, 10792, …} A natural number that is abundant but not semiperfect.
A006842 Farey sequence numerators {0, 1, 0, 1, 1, 0, 1, 1, 2, 1, …}
A006843 Farey sequence denominators {1, 1, 1, 2, 1, 1, 3, 2, 3, 1, …}
A006862 Euclid numbers {2, 3, 7, 31, 211, 2311, 30031, 510511, 9699691, 223092871, …} pn# + 1, i.e. 1 + product of first n consecutive primes.
A006886 Kaprekar numbers {1, 9, 45, 55, 99, 297, 703, 999, 2223, 2728, …} X2 = Abn + B, where 0 < B < bn and X = A + B.
A007304 Sphenic numbers {30, 42, 66, 70, 78, 102, 105, 110, 114, 130, …} Products of 3 distinct primes.
A007318 Pascal's triangle {1, 1, 1, 1, 2, 1, 1, 3, 3, 1, …} Pascal's triangle read by rows.
A007540 Wilson primes {5, 13, 563} Primes ${\displaystyle p}$ satisfying (p-1)! ≡ -1 (mod p2).
A007588 Stella octangula numbers {0, 1, 14, 51, 124, 245, 426, 679, 1016, 1449, 1990, 2651, 3444, 4381, …} Stella octangula numbers: n (2n2 − 1), with n ≥ 0.
A007770 Happy numbers {1, 7, 10, 13, 19, 23, 28, 31, 32, 44, …} The numbers whose trajectory under iteration of sum of squares of digits map includes 1.
A007947 Radical of an integer {1, 2, 3, 2, 5, 6, 7, 2, 3, 10, …} The radical of a positive integer n is the product of the distinct prime numbers dividing n.
A010060 Prouhet–Thue–Morse constant {0, 1, 1, 0, 1, 0, 0, 1, 1, 0, …} ${\displaystyle \tau =\sum _{i=0}^{\infty }{\frac {t_{i}}{2^{i+1}}}.}$
A014080 Factorions {1, 2, 145, 40585, …} A natural number that equals the sum of the factorials of its decimal digits.
A014577 Regular paperfolding sequence {1, 1, 0, 1, 1, 0, 0, 1, 1, 1, …} At each stage an alternating sequence of 1s and 0s is inserted between the terms of the previous sequence.
A016114 Circular primes {2, 3, 5, 7, 11, 13, 17, 37, 79, 113, …} The numbers which remain prime under cyclic shifts of digits.
A018226 Magic numbers {2, 8, 20, 28, 50, 82, 126, …} A number of nucleons (either protons or neutrons) such that they are arranged into complete shells within the atomic nucleus.
A019279 Superperfect numbers {2, 4, 16, 64, 4096, 65536, 262144, 1073741824, 1152921504606846976, 309485009821345068724781056, …} Positive integers n for which σ2(n) = σ(σ(n)) = 2n.
A027641 Bernoulli numbers Bn {1, -1, 1, 0, -1, 0, 1, 0, -1, 0, 5, 0, -691, 0, 7, 0, -3617, 0, 43867, 0, …}
A031214 First elements in all OEIS sequences {1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, …} One of sequences referring to the OEIS itself.
A033307 Decimal expansion of Champernowne constant {1, 2, 3, 4, 5, 6, 7, 8, 9, 1, …} Formed by concatenating the positive integers.
A034897 Hyperperfect numbers {6, 21, 28, 301, 325, 496, 697, …} k-hyperperfect numbers, i.e. n for which the equality n = 1 + k (σ(n) − n − 1) holds.
A035513 Wythoff array {1, 2, 4, 3, 7, 6, 5, 11, 10, 9, …} A matrix of integers derived from the Fibonacci sequence.
A036262 Gilbreath's conjecture {2, 1, 3, 1, 2, 5, 1, 0, 2, 7, …} Triangle of numbers arising from Gilbreath's conjecture.
A037274 Home prime {1, 2, 3, 211, 5, 23, 7, 3331113965338635107, 311, 773, …} For n ≥ 2, a(n) is the prime that is finally reached when you start with n, concatenate its prime factors (A037276) and repeat until a prime is reached; a(n) = − 1 if no prime is ever reached.
A046075 Undulating numbers {101, 121, 131, 141, 151, 161, 171, 181, 191, 202, …} A number that has the digit form ababab.
A050278 Pandigital numbers {1023456789, 1023456798, 1023456879, 1023456897, 1023456978, 1023456987, 1023457689, 1023457698, 1023457869, 1023457896, …} Numbers containing the digits 0–9 such that each digit appears exactly once.
A052486 Achilles numbers {72, 108, 200, 288, 392, 432, 500, 648, 675, 800, …} Positive integers which are powerful but imperfect.
A060006 Decimal expansion of Pisot–Vijayaraghavan number {1, 3, 2, 4, 7, 1, 7, 9, 5, 7, …} Real root of x3x − 1.
A076336 Sierpinski numbers {78557, 271129, 271577, 322523, 327739, 482719, 575041, 603713, 903983, 934909, …} Odd k for which { k⋅2n + 1 : n ∈ ℕ } consists only of composite numbers.
A076337 Riesel numbers {509203, 762701, 777149, 790841, 992077, …} Odd k for which { k⋅2n − 1 : n ∈ ℕ } consists only of composite numbers.
A086747 Baum–Sweet sequence {1, 1, 0, 1, 1, 0, 0, 1, 0, 1, …} a(n) = 1 if the binary representation of n contains no block of consecutive zeros of odd length; otherwise a(n) = 0.
A088164 Wolstenholme primes {16843, 2124679} Primes ${\displaystyle p}$ satisfying ${\displaystyle {2p-1 \choose p-1}\equiv 1{\pmod {p^{4}}}}$.
A090822 Gijswijt's sequence {1, 1, 2, 1, 1, 2, 2, 2, 3, 1, …} The nth term counts the maximal number of repeated blocks at the end of the subsequence from 1 to n-1
A094683 Juggler sequence {0, 1, 1, 5, 2, 11, 2, 18, 2, 27, …} If n ≡ 0 (mod 2) then n else n3/2.
A097942 Highly totient numbers {1, 2, 4, 8, 12, 24, 48, 72, 144, 240, …} Each number k on this list has more solutions to the equation φ(x) = k than any preceding k.
A100264 Decimal expansion of Chaitin's constant {0, 0, 7, 8, 7, 4, 9, 9, 6, 9, …} Chaitin constant (Chaitin omega number) or halting probability.
A104272 Ramanujan primes {2, 11, 17, 29, 41, 47, 59, 67, …} The nth Ramanujan prime is the least integer Rn for which π(x) − π(x/2) ≥ n, for all xRn.
A122045 Euler numbers {1, 0, −1, 0, 5, 0, −61, 0, 1385, 0, …} ${\displaystyle {\frac {1}{\cosh t}}={\frac {2}{e^{t}+e^{-t}}}=\sum _{n=0}^{\infty }{\frac {E_{n}}{n!}}\cdot t^{n}.}$
A138591 Polite numbers {3, 5, 6, 7, 9, 10, 11, 12, 13, 14, 15, 17, …} A positive integer that can be written as the sum of two or more consecutive positive integers.