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

1,1

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

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.

Links

Table of n, a(n) for n=1..46.

Example

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.

Mathematica

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

Crossrefs

Cf. A051161.

Sequence in context: A111892 A108248 A087104 * A069926 A077429 A204553

Adjacent sequences: A343740 A343741 A343742 * A343744 A343745 A343746

Keyword

nonn,fini,full

Author

Hal M. Switkay, Jun 27 2021

Status

approved