240 (number) explained

Number:	240
Divisor:	1, 2, 3, 4, 5, 6, 8, 10, 12, 15, 16, 20, 24, 30, 40, 48, 60, 80, 120, 240

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

Mathematics

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 the 12th highly composite number,^[4] with 20 divisors in total, more than any smaller number;^[5] and a refactorable number or tau number, since one of its divisors is 20, which divides 240 evenly.^[6]

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:^[7]

240 is highly totient, since it has thirty-one totient answers, more than any previous integer.^[8]

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,

r _{16} . ^{[9] [10] [11]}

240 is the number of distinct solutions of the Soma cube puzzle.^[12]

There are exactly 240 visible pieces of what would be a four-dimensional version of the Rubik's Revenge — a

4 x 4 x 4 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.[ 5 , 3 , 2 ] , of order 240; four-dimensional icosahedral prisms with Weyl group H _{3} x A _{1} 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

A _{1} x A _{4} is of order 240, where its largest member, the omnitruncated 5-cell prism, contains 240 edges. In the still five-dimensional H _{4} x A _{1} 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.

Notes and References

1. 2016-05-30.
2. Web site: Sloane's A005835 : Pseudoperfect (or semiperfect) numbers. The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. 2016-09-05.
3. Web site: Sloane's A050480 : Numbers that can be written as a concatenation of distinct proper divisors. The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. 2016-09-05.
4. Web site: A002182 - OEIS . 2024-11-28 . oeis.org.
5. Web site: Sloane's A002182 : Highly composite numbers . 2016-05-31 . The On-Line Encyclopedia of Integer Sequences . OEIS Foundation.
6. 2016-04-18.
7. Web site: 2009-08-15 . Sloane's A067373 : Integers expressible as the sum of (at least two) consecutive primes in at least 3 ways . 2021-08-27 . The On-Line Encyclopedia of Integer Sequences. . OEIS Foundation.
8. Web site: Sloane's A097942 : Highly totient numbers . 2016-05-28 . The On-Line Encyclopedia of Integer Sequences . OEIS Foundation.
9. Bailey . D. H. . Borwein . J. M. . Kapoor . V. . Weisstein . E. W. . Ten Problems in Experimental Mathematics. . . 113 . 6 . . 2006 . 482–485 . 10.2307/27641975 . 27641975 . 2231135 . 1153.65301 . 13560576 .
10. 2024-02-29 .
11. 2024-02-29 .
12. Web site: Weisstein. Eric W.. Soma Cube. Wolfram MathWorld. 2016-09-05.