240 (number)

← 239 240 241 →
← 240 241 242 243 244 245 246 247 248 249 → List of numbers Integers ← 0 100 200 300 400 500 600 700 800 900 →
Cardinal	two hundred forty
Ordinal	240th (two hundred fortieth)
Factorization	24 × 3 × 5
Divisors	1, 2, 3, 4, 5, 6, 8, 10, 12, 15, 16, 20, 24, 30, 40, 48, 60, 80, 120, 240
Greek numeral	ΣΜ´
Roman numeral	CCXL
Binary	111100002
Ternary	222203
Senary	10406
Octal	3608
Duodecimal	18012
Hexadecimal	F016

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 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.

References

1. Sloane, N. J. A. (ed.). "SequenceA002378(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.). "SequenceA033950(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.). "SequenceA091517(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.). "SequenceA118454(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.