cp's OEIS Frontend

A336940 Number of odd divisors of n!.

%I A336940 #28 Aug 27 2020 10:09:17
%S A336940 1,1,1,2,2,4,6,12,12,20,30,60,72,144,216,336,336,672,864,1728,2160,
%T A336940 3200,4800,9600,10560,14784,22176,28224,35280,70560,86400,172800,
%U A336940 172800,245760,368640,497664,559872,1119744,1679616,2363904,2626560,5253120,6451200,12902400,16128000
%N A336940 Number of odd divisors of n!.
%H A336940 Seiichi Manyama, <a href="/A336940/b336940.txt">Table of n, a(n) for n = 0..10000</a>
%F A336940 a(n) = A001227(n!).
%F A336940 a(n) = A000005(A049606(n)).
%F A336940 a(n) + A337257(n) = A027423(n) = A000005(n!).
%F A336940 From _Seiichi Manyama_, Aug 27 2020: (Start)
%F A336940 If p is odd prime, a(p) = 2 * a(p-1).
%F A336940 a(n) = A027423(n) / A113474(n) for n > 0. (End)
%e A336940 The a(1) = 1 through a(8) = 12 divisors:
%e A336940   1  1  1  1  1   1   1    1
%e A336940         3  3  3   3   3    3
%e A336940               5   5   5    5
%e A336940               15  9   7    7
%e A336940                   15  9    9
%e A336940                   45  15   15
%e A336940                       21   21
%e A336940                       35   35
%e A336940                       45   45
%e A336940                       63   63
%e A336940                       105  105
%e A336940                       315  315
%t A336940 Table[Length[Select[Divisors[n!],OddQ]],{n,0,15}]
%o A336940 (PARI) a(n) = sumdiv(n!, d, d%2); \\ _Michel Marcus_, Aug 24 2020
%o A336940 (PARI) a(n) = numdiv(prod(k=1, n, k >> valuation(k, 2))); \\ _Michel Marcus_, Aug 27 2020
%Y A336940 A049606 gives the maximum among these divisors, with quotient A060818.
%Y A336940 A337257 is the even version.
%Y A336940 A000265 gives the maximum odd divisor of n.
%Y A336940 A001227 counts odd divisors.
%Y A336940 A183063 counts even divisors.
%Y A336940 Cf. A000005, A001013, A001055, A006939, A113474, A124010, A253249.
%Y A336940 Factorial numbers: A000142, A022559, A027423 (divisors), A048656, A071626, A076716 (factorizations), A325272, A325273, A325617, A336414, A336498.
%K A336940 nonn
%O A336940 0,4
%A A336940 _Gus Wiseman_, Aug 23 2020
%E A336940 a(36)-a(44) from _Seiichi Manyama_, Aug 26 2020