240 (number)

From Wikipedia, the free encyclopedia
← 239 240 241 →
Cardinaltwo hundred forty
Ordinal240th
(two hundred fortieth)
Factorization24 × 3 × 5
Divisors1, 2, 3, 4, 5, 6, 8, 10, 12, 15, 16, 20, 24, 30, 40, 48, 60, 80, 120, 240
Greek numeralΣΜ´
Roman numeralCCXL
Binary111100002
Ternary222203
Senary10406
Octal3608
Duodecimal18012
HexadecimalF016

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

Mathematics[edit]

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.

E8 has 240 roots.

References[edit]

  1. ^ Sloane, N. J. A. (ed.). "Sequence A002378 (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.). "Sequence A033950 (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 – via JSTOR.
  9. ^ Sloane, N. J. A. (ed.). "Sequence A091517 (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.). "Sequence A118454 (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.
Retrieved from "https://en.wikipedia.org/w/index.php?title=240_(number)&oldid=1217197356"