72

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72 is an integer, the maximum number of spheres that can touch another sphere in a lattice packing in 6 dimensions.

Membership in core sequences

Even numbers	..., 66, 68, 70, 72, 74, 76, 78, ...	A005843
Composite numbers	..., 68, 69, 70, 72, 74, 75, 76, ...	A002808
Abundant numbers	..., 60, 66, 70, 72, 78, 80, 84, ...	A005101
Oblong numbers	..., 30, 42, 56, 72, 90, 110, 132, ...	A002378
Quarter-squares	..., 49, 56, 64, 72, 81, 90, 100, ...	A002620
Sums of two squares	..., 64, 65, 68, 72, 73, 74, 80, ...	A001481

Sequences pertaining to 72

Multiples of 72	0, 72, 144, 216, 288, 360, 432, 504, 576, 648, 720, 792, 864, ...
Divisors of 72	1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36, 72	A018271
$3x+1$ sequence beginning at 72	72, 36, 18, 9, 28, 14, 7, 22, 11, 34, 17, 52, 26, 13, 40, 20, ...

Partitions of 72

There are 5392783 partitions of 72.

The Goldbach representations of 72 are: 67 + 5 = 61 + 11 = 59 + 13 = 53 + 19 = 43 + 29 = 41 + 31.

Roots and powers of 72

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

TABLE GOES HERE

Values for number theoretic functions with 72 as an argument

$\mu (72)$	0
$M(72)$	−3
$\pi (72)$	20
$\sigma _{1}(72)$	195
$\sigma _{0}(72)$	12
$\phi (72)$	24
$\Omega (72)$	5
$\omega (72)$	2
$\lambda (72)$	6	This is the Carmichael lambda function.
$\lambda (72)$	−1	This is the Liouville lambda function.
72!	Approx. 6.1234458376886 × 10 103
$\Gamma (72)$	Approx. 8.5047858856786 × 10 101

Factorization of 72 in some quadratic integer rings

As was mentioned above, 72 is the product of the cube of a prime and the square of a prime in $\mathbb {Z}$. But it has different factorizations in some quadratic integer rings.

TABLE

REMARKS

Representation of 72 in various bases

Base	2	3	4	5	6	7	8	9	10	11	12	13	14	15	16	17	18	19	20
Representation	1001000	2200	1020	242	200	132	110	80	72	66	60	57	52	4C	48	44	40	3F	3C

72 is a Harshad number in every base from binary to base 13.

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