240 (number)

Template:Infobox 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 the 12th 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: $$\begin{array}{cc}240& =113+127\\ 240& =53+59+61+67\\ 240& =17+19+23+29+31+37+41+43\\ \end{array}$$

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 ${\mathrm{H}}_{\mathrm{3}}\times {\mathrm{A}}_{\mathrm{1}}$ 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 ${\mathrm{A}}_{\mathrm{1}}\times {\mathrm{A}}_{\mathrm{4}}$ is of order 240, where its largest member, the omnitruncated 5-cell prism, contains 240 edges. In the still five-dimensional ${\mathrm{H}}_{\mathrm{4}}\times {\mathrm{A}}_{\mathrm{1}}$ 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.