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A226143 a(n) is the smallest k > 0 such that A000793(n)^k >= n!.

%I A226143 #26 Feb 23 2025 02:00:07
%S A226143 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 A226143 10,10,11,11,11,11,11,11,12,12,12,12,12,13,12,13,12,13,13,13,13,13,14,
%U A226143 14,14,14,14,14,14,15,15,15,15,15,15,15,16,15,16,16
%N A226143 a(n) is the smallest k > 0 such that A000793(n)^k >= n!.
%C A226143 This is a lower bound for A226142(n), the least positive integer k such that S_n is a product of k cyclic groups.
%H A226143 Alois P. Heinz, <a href="/A226143/b226143.txt">Table of n, a(n) for n = 1..10000</a>
%F A226143 a(n) = ceiling(log_m(n!)) where m = A000793(n).
%p A226143 A000793:=
%p A226143 [1,2,3,4,6,6,12,15,20,30,30,60,60,84,105,140,
%p A226143 210,210,420,420,420,420,840,840,1260,1260,1540,
%p A226143 2310,2520,4620,4620,5460,5460,9240,9240,13860,
%p A226143 13860,16380,16380,27720,30030,32760,60060,60060,
%p A226143 60060,60060,120120]:
%p A226143 a:=proc(n)
%p A226143 global A000793;
%p A226143 local k;
%p A226143 for k from 1 do
%p A226143 if A000793[n]^k >= n! then return k; fi;
%p A226143 od;
%p A226143 end;
%t A226143 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 A226143 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 A226143 Array[a, 100] (* _Jean-François Alcover_, Nov 12 2020, after _Alois P. Heinz_ in A000793 *)
%Y A226143 Cf. A000793, A226142.
%K A226143 nonn
%O A226143 1,3
%A A226143 _W. Edwin Clark_, May 27 2013