240 (number)

Short description: Natural number

| <- 239  | 240 |  241 ->

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← 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
Quaternary| 33004
Quinary| 14305
Senary| 10406
Octal| 3608
Duodecimal| 18012
Hexadecimal| F016
Vigesimal| C020
Base 36| 6O36

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 twenty divisors total, more than any previous 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] [math]\displaystyle{ \begin{align} 240 & = 113 + 127 \\\ 240 & = 53 + 59 + 61 + 67 \\\ 240 & = 17 + 19 + 23 + 29 + 31 + 37 + 41 + 43 \\\ \end{align} }[/math]

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 number of distinct solutions of the Soma cube puzzle.[8]

There are exactly 240 visible pieces of what would be a four-dimensional version of the Rubik's Revenge — a [math]\displaystyle{ 4 \times 4 \times 4 }[/math] 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

1. ↑ Sloane, N. J. A., ed. "Sequence A002378 (Oblong (or promic, pronic, or heteromecic) numbers)". OEIS Foundation. https://oeis.org/A002378. Retrieved 2016-05-30.
2. ↑ "Sloane's A005835 : Pseudoperfect (or semiperfect) numbers". OEIS Foundation. https://oeis.org/A005835.
3. ↑ "Sloane's A050480 : Numbers that can be written as a concatenation of distinct proper divisors". OEIS Foundation. https://oeis.org/A050480.
4. ↑ "Sloane's A002182 : Highly composite numbers". OEIS Foundation. https://oeis.org/A002182.
5. ↑ Sloane, N. J. A., ed. "Sequence A033950 (Refactorable numbers)". OEIS Foundation. https://oeis.org/A033950. Retrieved 2016-04-18.
6. ↑ "Sloane's A067373 : Integers expressible as the sum of (at least two) consecutive primes in at least 3 ways". OEIS Foundation. 2009-08-15. https://oeis.org/A067373.
7. ↑ "Sloane's A097942 : Highly totient numbers". OEIS Foundation. https://oeis.org/A097942.
8. ↑ Weisstein, Eric W.. "Soma Cube". http://mathworld.wolfram.com/SomaCube.html.

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