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

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, 10, 10, 11, 11, 11, 11, 11, 11, 12, 12, 12, 12, 12, 13, 12, 13, 12, 13, 13, 13, 13, 13, 14, 14, 14, 14, 14, 14, 14, 15, 15, 15, 15, 15, 15, 15, 16, 15, 16, 16

Offset: 1

Author

W. Edwin Clark, May 27 2013

Keywords

• nonn

Comments

This is a lower bound for A226142(n), the least positive integer k such that S_n is a product of k cyclic groups.

Links

• Alois P. Heinz, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A000793, A226142.

Programs

Maple

A000793:=
[1,2,3,4,6,6,12,15,20,30,30,60,60,84,105,140,
210,210,420,420,420,420,840,840,1260,1260,1540,
2310,2520,4620,4620,5460,5460,9240,9240,13860,
13860,16380,16380,27720,30030,32760,60060,60060,
60060,60060,120120]:
a:=proc(n)
global A000793;
local k;
for k from 1 do
if A000793[n]^k >= n! then return k; fi;
od;
end;

Mathematica

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}]]]];
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]];
Array[a, 100] (* Jean-François Alcover, Nov 12 2020, after Alois P. Heinz in A000793 *)

Formula

a(n) = ceiling(log_m(n!)) where m = A000793(n).