A343743 a(n) is the largest base in which the order of the Monster group has (47 - n) zeros; alternatively, radicals of maximal powers dividing the order of the Monster group.

2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 4, 4, 4, 12, 12, 12, 12, 12, 24, 24, 24, 24, 48, 144, 1440, 1440, 2880, 120960, 1451520, 87091200, 1902071808000, 15184923989114880000, 808017424794512875886459904961710757005754368000000000

Offset: 1

Author

Hal M. Switkay, Jun 27 2021

Keywords

• nonn
• fini
• full

Comments

Let z be a specified minimum number of zeros in the order of the Monster group; here z is a natural number, 1 <= z <= 46, with z = (47 - n). Then the largest base in which the order of the Monster group has at least z zeros is:

Product_{k=1..20} prime(k)^floor(A051161(k)/z).

When z = 1 this is the order of the Monster group.

Every term in this sequence except the last is a number of least prime signature (A025487).

In the following table, when the order of the Monster group has exactly z zeros, it also has s significant digits, and d = s + z total digits.

z s d

-- --- ---

46 134 180

23 67 90

20 30 50

15 25 40

11 22 33

10 15 25

9 9 18

7 9 16

6 5 11

5 4 9

4 3 7

3 2 5

2 1 3

1 1 2

Examples

			a(27) = the largest base in which the order of the Monster group has at least (47 - 27) = 20 zeros. This is 2^(floor(46/20)) * 3^(floor(20/20)) = 2^2 * 3 = 12; the remaining terms in the product have exponent 0.
		

References

• J. H. Conway, R. T. Curtis, S. P. Norton, R. A. Parker and R. A. Wilson, ATLAS of Finite Groups. Oxford Univ. Press, 1985 [for best online version see https://oeis.org/wiki/Welcome#Links_to_Other_Sites].
• J. H. Conway, N. J. A. Sloane, Sphere Packings, Lattices, and Groups. Springer, 3rd ed., 1999.

Crossrefs

Cf. A051161.

Programs

Mathematica

f = FactorInteger[MonsterGroupM[] // GroupOrder]; Table[Times @@ ((First[#]^Floor[Last[#]/z]) & /@ f), {z, Max[f[[;; , 2]]], 1, -1}] (* Amiram Eldar, Jul 19 2021 *)