A211392 The number of divisors d of n! such that the symmetric group on n letters contains no elements of order d.

0, 0, 1, 4, 10, 24, 51, 85, 146, 254, 520, 769, 1557, 2561, 3997, 5333, 10705, 14633, 29315, 40970, 60722, 95912, 191902, 242769, 339909, 532088, 677224, 917112, 1834373, 2332596, 4665375, 5529352, 7864049, 12164824, 16422587, 19595164, 39190653, 60465758

Offset: 1

Author

Alexander Gruber, Feb 07 2013

Keywords

• nonn

Links

• Alois P. Heinz, Table of n, a(n) for n = 1..1000

Crossrefs

Cf. A000005, A009490, A027423, A211391.

Programs

Maple

b:= proc(n,i) option remember; local p;
      p:= `if`(i<1, 1, ithprime(i));
      `if`(n=0 or i<1, 1, b(n, i-1)+
      add(b(n-p^j, i-1), j=1..ilog[p](n)))
    end:
a:= n-> numtheory[tau](n!) -b(n, numtheory[pi](n)):
seq(a(n), n=1..100);  # Alois P. Heinz, Feb 15 2013

Mathematica

b[n_, i_] := b[n, i] = Module[{p}, p = If[i<1, 1, Prime[i]]; If[n==0 || i<1, 1, b[n, i-1] + Sum[b[n-p^j, i-1], {j, 1, Floor@Log[p, n]}]]];
a[n_] := DivisorSigma[0, n!] - b[n, PrimePi[n]];
Table[a[n], {n, 1, 100}] (* Jean-François Alcover, Mar 24 2017, after Alois P. Heinz *)

Formula

a(n) = A000005(n!) - A009490(n).

Extensions

More terms from Alois P. Heinz, Feb 11 2013