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# 120

Please do not rely on any information it contains.

120 is an integer. It is believed to be the largest triangular number among the factorials.

## Membership in core sequences

 Even numbers ..., 114, 116, 118, 120, 122, 124, 126, ... A005843 Composite numbers ..., 117, 118, 119, 120, 121, 122, 123, ... A002808 Factorials 1, 1, 2, 6, 24, 120, 720, 5040, 40320, ... A000142 Abundant numbers ..., 108, 112, 114, 120, 126, 132, 138, ... A005101 Triangular numbers ..., 78, 91, 105, 120, 136, 153, 171, ... A000217 Tetrahedral numbers ..., 35, 56, 84, 120, 165, 220, 286, 364, ... A000292

In Pascal's triangle, 120 occurs six times, the first two times in row 10 as the sum of 36 and 84 from row 9.

## Sequences pertaining to 120

 Divisors of 120 1, 2, 3, 4, 5, 6, 8, 10, 12, 15, 20, 24, 30, 40, 60, 120 A018293 Multiples of 120 120, 240, 360, 480, 600, 720, 840, 960, 1080, 1200, 1320, 1440, ... ${\displaystyle 3x+1}$ sequence starting at 120 120, 60, 30, 15, 46, 23, 70, 35, 106, 53, 160, 80, 40, 20, 10, 5, 16, ...

## Partitions of 120

There are 1844349560 partitions of 120.

The Goldbach representations of 120 are: 113 + 7 = 109 + 11 = 107 + 13 = 103 + 17 = 101 + 19 = 97 + 23 = 89 + 31 = 83 + 37 = 79 + 41 = 73 + 47 = 67 + 53 = 61 + 59 = 120.

## Roots and powers of 120

In the table below, irrational numbers are given truncated to eight decimal places.

TABLE GOES HERE

## Logarithms and 120th powers

In the OEIS specifically and mathematics in general, ${\displaystyle \log x}$ refers to the natural logarithm of ${\displaystyle x}$, whereas all other bases are specified with a subscript.

As above, irrational numbers in the following table are truncated to eight decimal places.

TABLE GOES HERE

## Values for number theoretic functions with 120 as an argument

 ${\displaystyle \mu (120)}$ 0 ${\displaystyle M(120)}$ −3 ${\displaystyle \pi (120)}$ 30 ${\displaystyle \sigma _{1}(120)}$ 360 Note that this is thrice 120.See A005820. ${\displaystyle \sigma _{0}(120)}$ 16 ${\displaystyle \phi (120)}$ 32 ${\displaystyle \Omega (120)}$ 5 ${\displaystyle \omega (120)}$ 3 ${\displaystyle \lambda (120)}$ 4 This is the Carmichael lambda function. ${\displaystyle \lambda (120)}$ −1 This is the Liouville lambda function. 120! Aprroximately 6.68950291 × 10 198 ${\displaystyle \Gamma (120)}$ Aprroximately 5.57458576 × 10 196

## Factorization of 120 in some quadratic integer rings

As was mentioned above, 120 is the product of 2 3, 3 and 5. But it has different factorizations in some quadratic integer rings.

 ${\displaystyle \mathbb {Z} [i]}$ ${\displaystyle (1\pm i)^{3}3(2\pm i)}$ ${\displaystyle \mathbb {Z} [{\sqrt {-2}}]}$ ${\displaystyle (-1)({\sqrt {-2}})^{6}(1\pm {\sqrt {-2}})5}$ ${\displaystyle \mathbb {Z} [{\sqrt {2}}]}$ ${\displaystyle ({\sqrt {2}})^{6}\times 3\times 5}$ ${\displaystyle \mathbb {Z} [\omega ]}$ ${\displaystyle (-1)2^{3}(1+2\omega )^{2}5}$ ${\displaystyle \mathbb {Z} [{\sqrt {3}}]}$ ${\displaystyle (1\pm {\sqrt {3}})^{3}({\sqrt {3}})^{2}5}$ ${\displaystyle \mathbb {Z} [{\sqrt {-5}}]}$ ${\displaystyle (-1)\times 2^{3}\times 3\times ({\sqrt {-5}})^{2}}$ ${\displaystyle \mathbb {Z} [\phi ]}$ ${\displaystyle 2^{3}\times 3\times (-1+2\phi )^{2}}$ ${\displaystyle \mathbb {Z} [{\sqrt {-6}}]}$ 2 3 × 3 × 5 OR ${\displaystyle (-1)2({\sqrt {-6}})^{2}5}$ ${\displaystyle \mathbb {Z} [{\sqrt {6}}]}$ ${\displaystyle (2\pm {\sqrt {6}})^{3}(3\pm {\sqrt {6}})(1\pm {\sqrt {6}})}$ ${\displaystyle {\mathcal {O}}_{\mathbb {Q} ({\sqrt {-7}})}}$ ${\displaystyle \left({\frac {1}{2}}\pm {\frac {\sqrt {-7}}{2}}\right)^{3}\times 3\times 5}$ ${\displaystyle \mathbb {Z} [{\sqrt {7}}]}$ ${\displaystyle (-1)(3\pm {\sqrt {7}})^{3}(2\pm {\sqrt {7}})5}$ ${\displaystyle \mathbb {Z} [{\sqrt {-10}}]}$ 2 3 × 3 × 5 ${\displaystyle \mathbb {Z} [{\sqrt {10}}]}$ 2 3 × 3 × 5 OR ${\displaystyle 2^{2}(4\pm {\sqrt {10}})5}$ ${\displaystyle {\mathcal {O}}_{\mathbb {Q} ({\sqrt {-11}})}}$ ${\displaystyle 2^{3}\left({\frac {1}{2}}\pm {\frac {\sqrt {-11}}{2}}\right)\left({\frac {3}{2}}\pm {\frac {\sqrt {-11}}{2}}\right)}$ ${\displaystyle \mathbb {Z} [{\sqrt {11}}]}$ ${\displaystyle (3\pm {\sqrt {11}})^{3}3(4\pm {\sqrt {11}})}$ ${\displaystyle \mathbb {Z} [{\sqrt {-13}}]}$ 2 3 × 3 × 5 ${\displaystyle {\mathcal {O}}_{\mathbb {Q} ({\sqrt {13}})}}$ ${\displaystyle (-1)2^{3}\left({\frac {1}{2}}\pm {\frac {\sqrt {13}}{2}}\right)5}$ ${\displaystyle \mathbb {Z} [{\sqrt {-14}}]}$ ${\displaystyle \mathbb {Z} [{\sqrt {14}}]}$ ${\displaystyle (-1)(4\pm {\sqrt {14}})^{3}3(3\pm {\sqrt {14}})}$ ${\displaystyle {\mathcal {O}}_{\mathbb {Q} ({\sqrt {-15}})}}$ 2 3 × 3 × 5 OR ${\displaystyle \left({\frac {1}{2}}\pm {\frac {\sqrt {-15}}{2}}\right)\times 3\times 5}$ ${\displaystyle \mathbb {Z} [{\sqrt {15}}]}$ 2 3 × 3 × 5 OR ${\displaystyle (-1)2^{2}(3\pm {\sqrt {15}})5}$ ${\displaystyle \mathbb {Z} [{\sqrt {-17}}]}$ 2 3 × 3 × 5 ${\displaystyle {\mathcal {O}}_{\mathbb {Q} ({\sqrt {17}})}}$ ${\displaystyle \left({\frac {3}{2}}\pm {\frac {\sqrt {17}}{2}}\right)^{3}\times 3\times 5}$ ${\displaystyle {\mathcal {O}}_{\mathbb {Q} ({\sqrt {-19}})}}$ ${\displaystyle 2\times 3\times \left({\frac {1}{2}}-{\frac {\sqrt {-19}}{2}}\right)\left({\frac {1}{2}}+{\frac {\sqrt {-19}}{2}}\right)}$ ${\displaystyle \mathbb {Z} [{\sqrt {19}}]}$ ${\displaystyle (-1)(13\pm 3{\sqrt {19}})^{2}(4\pm {\sqrt {19}})(9\pm 2{\sqrt {19}})}$

## Representation of 120 in various bases

 Base 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 Representation 1111000 11110 1320 440 320 231 170 143 120 AA A0 93 88 80 78 71 6C 66 60

 ${\displaystyle -1}$ 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 1729