A211392 - OEIS

%I #25 Nov 22 2023 21:38:42

%S 0,0,1,4,10,24,51,85,146,254,520,769,1557,2561,3997,5333,10705,14633,

%T 29315,40970,60722,95912,191902,242769,339909,532088,677224,917112,

%U 1834373,2332596,4665375,5529352,7864049,12164824,16422587,19595164,39190653,60465758

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

%H Alois P. Heinz, <a href="/A211392/b211392.txt">Table of n, a(n) for n = 1..1000</a>

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

%p b:= proc(n,i) option remember; local p;

%p       p:= `if`(i<1, 1, ithprime(i));

%p       `if`(n=0 or i<1, 1, b(n, i-1)+

%p       add(b(n-p^j, i-1), j=1..ilog[p](n)))

%p     end:

%p a:= n-> numtheory[tau](n!) -b(n, numtheory[pi](n)):

%p seq(a(n), n=1..100);  # _Alois P. Heinz_, Feb 15 2013

%t 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]}]]];

%t a[n_] := DivisorSigma[0, n!] - b[n, PrimePi[n]];

%t Table[a[n], {n, 1, 100}] (* _Jean-François Alcover_, Mar 24 2017, after _Alois P. Heinz_ *)

%Y Cf. A000005, A009490, A027423, A211391.

%K nonn

%O 1,4

%A _Alexander Gruber_, Feb 07 2013

%E More terms from _Alois P. Heinz_, Feb 11 2013