240 (number)

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. It is a semiperfect number, equal to the concatenation of two of its proper divisors (24 and 40).

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

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

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

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, $$r_{16}.$$

240 is the number of distinct solutions of the Soma cube puzzle.

There are exactly 240 visible pieces of what would be a four-dimensional version of the Rubik's Revenge — a $$4 \times 4 \times 4$$ 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 $$[5,3,2]$$, of order 240; four-dimensional icosahedral prisms with Weyl group $$\mathrm{H_{3}} \times \mathrm{A_{1}}$$ 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 $$\mathrm {A_{1}} \times \mathrm {A_{4}}$$ is of order 240, where its largest member, the omnitruncated 5-cell prism, contains 240 edges. In the still five-dimensional $$\mathrm {H_{4}} \times \mathrm {A_{1}}$$ 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.