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A195208 Number of odd divisors of !n.

%I A195208 #22 Aug 02 2024 03:15:55
%S A195208 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,
%T A195208 384,32,64,144,16,8,36,128,16,80,24,16,96,32,16,96,4,16,448,128,64,24,
%U A195208 192,48,192,64,16,96,768,32,96,32,64
%N A195208 Number of odd divisors of !n.
%C A195208 !k is a subfactorial number (A000166).
%H A195208 Amiram Eldar, <a href="/A195208/b195208.txt">Table of n, a(n) for n = 0..81</a>
%F A195208 a(n) = A000005(A000265(A000166(n))). - _Robert Israel_, Aug 01 2024
%F A195208 a(n) = A001227(A000166(n)), for n != 1. - _Amiram Eldar_, Aug 02 2024
%e A195208 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.
%p A195208 A166 := proc(n) option remember; (n-1)*(procname(n-1)+procname(n-2)); end:
%p A195208 A166(0):= 1: A166(1):= 0:
%p A195208 f:= proc(n) local v; v:= A166(n); numtheory:-tau(v/2^padic:-ordp(v,2)) end proc:
%p A195208 map(f, [$1..60]); # _Robert Israel_, Aug 01 2024
%t A195208 f[n_] := Block[{d = Divisors[Subfactorial[n]]}, Count[OddQ[d], True]]; Table[f[n], {n, 0, 60}]
%o A195208 (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
%Y A195208 Cf. A000005, A000166, A000265, A001227, A195207, A195209, A195210.
%K A195208 nonn
%O A195208 0,5
%A A195208 _Michel Lagneau_, Sep 13 2011