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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.
Contents
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 |
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
0 | ||
−3 | ||
20 | ||
195 | ||
12 | ||
24 | ||
5 | ||
2 | ||
6 | This is the Carmichael lambda function. | |
−1 | This is the Liouville lambda function. | |
72! | Approx. 6.1234458376886 × 10 103 | |
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 . 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
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 |