240 (number)

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

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] $${\displaystyle {\begin{aligned}240&=113+127\\240&=53+59+61+67\\240&=17+19+23+29+31+37+41+43\\\end{aligned}}}$$

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, ${\displaystyle r_{16}.}$^[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 ${\displaystyle 4\times 4\times 4}$ 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.

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 ${\displaystyle [5,3,2]}$, of order 240; four-dimensional icosahedral prisms with Weyl group ${\displaystyle \mathrm {H_{3}} \times \mathrm {A_{1}} }$ 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 ${\displaystyle \mathrm {A_{1}} \times \mathrm {A_{4}} }$ is of order 240, where its largest member, the omnitruncated 5-cell prism, contains 240 edges. In the still five-dimensional ${\displaystyle \mathrm {H_{4}} \times \mathrm {A_{1}} }$ 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.