120

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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, ...
$3x+1$ 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, $\log x$ refers to the natural logarithm of $x$, 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

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

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

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

See also

Some integers

							$-1$
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