A195208 Number of odd divisors of !n.

1, 0, 1, 1, 3, 2, 4, 6, 8, 8, 10, 4, 6, 12, 8, 4, 24, 4, 8, 24, 32, 16, 24, 96, 32, 48, 24, 16, 384, 32, 64, 144, 16, 8, 36, 128, 16, 80, 24, 16, 96, 32, 16, 96, 4, 16, 448, 128, 64, 24, 192, 48, 192, 64, 16, 96, 768, 32, 96, 32, 64

Offset: 0

Author

Michel Lagneau, Sep 13 2011

Keywords

• nonn

Comments

!k is a subfactorial number (A000166).

Examples

			a(7) = 6 because the divisors of  !7 = 1854 are  {1, 2, 3, 6, 9, 18, 103, 206, 309, 618, 927, 1854} with 6  odd divisors  1, 3, 9, 103, 309 and 927 => 6 is in the sequence.
		

Links

• Amiram Eldar, Table of n, a(n) for n = 0..81

Crossrefs

Cf. A000005, A000166, A000265, A001227, A195207, A195209, A195210.

Programs

Maple

A166 := proc(n) option remember; (n-1)*(procname(n-1)+procname(n-2)); end:
A166(0):= 1: A166(1):= 0:
f:= proc(n) local v; v:= A166(n); numtheory:-tau(v/2^padic:-ordp(v,2)) end proc:
map(f, [$1..60]); # Robert Israel, Aug 01 2024

Mathematica

f[n_] := Block[{d = Divisors[Subfactorial[n]]}, Count[OddQ[d], True]]; Table[f[n], {n, 0, 60}]

PARI

a(n)=if(n<2,1-n,my(x='x,k=n!*polcoeff(exp(-x+x*O(x^n))/(1-x),n));numdiv(k>>valuation(k,2))) \\ Charles R Greathouse IV, Sep 13 2011

Formula

a(n) = A000005(A000265(A000166(n))). - Robert Israel, Aug 01 2024

a(n) = A001227(A000166(n)), for n != 1. - Amiram Eldar, Aug 02 2024