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# 72

Please do not rely on any information it contains.

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.

 $-1$ 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