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%I #20 Nov 12 2020 12:04:53
%S 1,1,2,3,3,4,4,4,5,5,6,5,6,6,6,7,7,7,7,8,8,9,8,9,9,9,9,9,10,9,10,10,
%T 10,10,11,11,11,11,11,11,12,12,12,12,12,13,12,13,12,13,13,13,13,13,14,
%U 14,14,14,14,14,14,15,15,15,15,15,15,15,16,15,16,16
%N a(n) is the smallest k > 0 such that A000793(n)^k >= n!.
%C This is a lower bound for A226142(n), the least positive integer k such that S_n is a product of k cyclic groups. Clearly also a(n) = ceiling(log_m(n!) where m = A000793(n).
%H Alois P. Heinz, <a href="/A226143/b226143.txt">Table of n, a(n) for n = 1..10000</a>
%p A000793:=
%p [1,2,3,4,6,6,12,15,20,30,30,60,60,84,105,140,
%p 210,210,420,420,420,420,840,840,1260,1260,1540,
%p 2310,2520,4620,4620,5460,5460,9240,9240,13860,
%p 13860,16380,16380,27720,30030,32760,60060,60060,
%p 60060,60060,120120]:
%p a:=proc(n)
%p global A000793;
%p local k;
%p for k from 1 do
%p if A000793[n]^k >= n! then return k; fi;
%p od;
%p end;
%t b[n_, i_] := b[n, i] = Module[{p}, p = If[i < 1, 1, Prime[i]]; If[n == 0 || i < 1, 1, Max[b[n, i - 1], Table[p^j b[n - p^j, i - 1], {j, 1, Log[p, n] // Floor}]]]];
%t a[n_] := Module[{m}, If[n == 1, 1, m = b[n, If[n < 8, 3, PrimePi[Ceiling[ 1.328 Sqrt[n Log[n] // Floor]]]]]; Log[m, n!] // Ceiling]];
%t Array[a, 100] (* _Jean-François Alcover_, Nov 12 2020, after _Alois P. Heinz_ in A000793 *)
%Y Cf. A000793, A226142.
%K nonn
%O 1,3
%A _W. Edwin Clark_, May 27 2013
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