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%I #76 Jun 14 2024 22:31:11
%S 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,
%T 24,24,24,24,48,144,1440,1440,2880,120960,1451520,87091200,
%U 1902071808000,15184923989114880000,808017424794512875886459904961710757005754368000000000
%N 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.
%C 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:
%C Product_{k=1..20} prime(k)^floor(A051161(k)/z).
%C When z = 1 this is the order of the Monster group.
%C Every term in this sequence except the last is a number of least prime signature (A025487).
%C 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.
%C z s d
%C -- --- ---
%C 46 134 180
%C 23 67 90
%C 20 30 50
%C 15 25 40
%C 11 22 33
%C 10 15 25
%C 9 9 18
%C 7 9 16
%C 6 5 11
%C 5 4 9
%C 4 3 7
%C 3 2 5
%C 2 1 3
%C 1 1 2
%D 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].
%D J. H. Conway, N. J. A. Sloane, Sphere Packings, Lattices, and Groups. Springer, 3rd ed., 1999.
%e 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.
%t f = FactorInteger[MonsterGroupM[] // GroupOrder]; Table[Times @@ ((First[#]^Floor[Last[#]/z]) & /@ f), {z, Max[f[[;; , 2]]], 1, -1}] (* _Amiram Eldar_, Jul 19 2021 *)
%Y Cf. A051161.
%K nonn,fini,full
%O 1,1
%A _Hal M. Switkay_, Jun 27 2021
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