240 (number)

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239 240 241
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]

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, [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 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 , of order 240; four-dimensional icosahedral prisms with Weyl group 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 is of order 240, where its largest member, the omnitruncated 5-cell prism, contains 240 edges. In the still five-dimensional 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.

Related Research Articles

90 (ninety) is the natural number following 89 and preceding 91.

72 (seventy-two) is the natural number following 71 and preceding 73. It is half a gross or six dozen.

36 (thirty-six) is the natural number following 35 and preceding 37.

58 (fifty-eight) is the natural number following 57 and preceding 59.

64 (sixty-four) is the natural number following 63 and preceding 65.

92 (ninety-two) is the natural number following 91 and preceding 93.

104 is the natural number following 103 and preceding 105.

100 or one hundred is the natural number following 99 and preceding 101.

1000 or one thousand is the natural number following 999 and preceding 1001. In most English-speaking countries, it can be written with or without a comma or sometimes a period separating the thousands digit: 1,000.

300 is the natural number following 299 and preceding 301.

180 is the natural number following 179 and preceding 181.

700 is the natural number following 699 and preceding 701.

800 is the natural number following 799 and preceding 801.

It is:

135 is the natural number following 134 and preceding 136.

100,000 (one hundred thousand) is the natural number following 99,999 and preceding 100,001. In scientific notation, it is written as 105.

168 is the natural number following 167 and preceding 169.

1728 is the natural number following 1727 and preceding 1729. It is a dozen gross, or one great gross. It is also the number of cubic inches in a cubic foot.

888 is the natural number following 887 and preceding 889.

14 (fourteen) is the natural number following 13 and preceding 15.

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.