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240 (number)

From Wikipedia, the free encyclopedia
← 239 240 241 →
Cardinaltwo hundred forty
Ordinal240th
(two hundred fortieth)
Factorization24 × 3 × 5
Divisors1, 2, 3, 4, 5, 6, 8, 10, 12, 15, 16, 20, 24, 30, 40, 48, 60, 80, 120, 240
Greek numeralΣΜ´
Roman numeralCCXL
Binary111100002
Ternary222203
Senary10406
Octal3608
Duodecimal18012
HexadecimalF016

240 (two hundred [and] forty) is the natural number following 239 and preceding 241.

Mathematics

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240 is a pronic number, since it can be expressed as the product of two consecutive integers, 15 and 16.[1] It is a semiperfect number,[2] equal to the concatenation of two of its proper divisors (24 and 40).[3]

It is also a highly composite number with 20 divisors in total, more than any smaller number;[4] and a refactorable number or tau number, since one of its divisors is 20, which divides 240 evenly.[5]

240 is the aliquot sum of only two numbers: 120 and 57121 (or 2392); and is part of the 12161-aliquot tree that goes: 120, 240, 504, 1056, 1968, 3240, 7650, 14112, 32571, 27333, 12161, 1, 0.

It is the smallest number that can be expressed as a sum of consecutive primes in three different ways:[6]

240 is highly totient, since it has thirty-one totient answers, more than any previous integer.[7]

It is palindromic in bases 19 (CC19), 23 (AA23), 29 (8829), 39 (6639), 47 (5547) and 59 (4459), while a Harshad number in bases 2, 3, 4, 5, 6, 7, 9, 10, 11, 13, 14, 15 (and 73 other bases).

240 is the algebraic polynomial degree of sixteen-cycle logistic map, [8][9][10]

240 is the number of distinct solutions of the Soma cube puzzle.[11]

There are exactly 240 visible pieces of what would be a four-dimensional version of the Rubik's Revenge — a Rubik's Cube. A Rubik's Revenge in three dimensions has 56 (64 – 8) visible pieces, which means a Rubik's Revenge in four dimensions has 240 (256 – 16) visible pieces.

240 is the number of elements in the four-dimensional 24-cell (or rectified 16-cell): 24 cells, 96 faces, 96 edges, and 24 vertices. On the other hand, the omnitruncated 24-cell, runcinated 24-cell, and runcitruncated 24-cell all have 240 cells, while the rectified 24-cell and truncated 24-cell have 240 faces. The runcinated 5-cell, bitruncated 5-cell, and omnitruncated 5-cell (the latter with 240 edges) all share pentachoric symmetry , of order 240; four-dimensional icosahedral prisms with Weyl group also have order 240. The rectified tesseract has 240 elements as well (24 cells, 88 faces, 96 edges, and 32 vertices).

In five dimensions, the rectified 5-orthoplex has 240 cells and edges, while the truncated 5-orthoplex and cantellated 5-orthoplex respectively have 240 cells and vertices. The uniform prismatic family is of order 240, where its largest member, the omnitruncated 5-cell prism, contains 240 edges. In the still five-dimensional prismatic group, the 600-cell prism contains 240 vertices. Meanwhile, in six dimensions, the 6-orthoplex has 240 tetrahedral cells, where the 6-cube contains 240 squares as faces (and a birectified 6-cube 240 vertices), with the 6-demicube having 240 edges.

E8 in eight dimensions has 240 roots.

References

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  1. ^ Sloane, N. J. A. (ed.). "Sequence A002378 (Oblong (or promic, pronic, or heteromecic) numbers)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-05-30.
  2. ^ "Sloane's A005835 : Pseudoperfect (or semiperfect) numbers". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-09-05.
  3. ^ "Sloane's A050480 : Numbers that can be written as a concatenation of distinct proper divisors". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-09-05.
  4. ^ "Sloane's A002182 : Highly composite numbers". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-05-31.
  5. ^ Sloane, N. J. A. (ed.). "Sequence A033950 (Refactorable numbers)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-04-18.
  6. ^ "Sloane's A067373 : Integers expressible as the sum of (at least two) consecutive primes in at least 3 ways". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. 2009-08-15. Retrieved 2021-08-27.
  7. ^ "Sloane's A097942 : Highly totient numbers". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-05-28.
  8. ^ Bailey, D. H.; Borwein, J. M.; Kapoor, V.; Weisstein, E. W. (2006). "Ten Problems in Experimental Mathematics" (PDF). American Mathematical Monthly. 113 (6). Taylor & Francis: 482–485. doi:10.2307/27641975. JSTOR 27641975. MR 2231135. S2CID 13560576. Zbl 1153.65301.
  9. ^ Sloane, N. J. A. (ed.). "Sequence A091517 (Decimal expansion of the value of r corresponding to the onset of the period 16-cycle in the logistic map.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2024-02-29.
  10. ^ Sloane, N. J. A. (ed.). "Sequence A118454 (Algebraic degree of the onset of the logistic map n-bifurcation.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2024-02-29.
  11. ^ Weisstein, Eric W. "Soma Cube". Wolfram MathWorld. Retrieved 2016-09-05.