A336940 Number of odd divisors of n!.

1, 1, 1, 2, 2, 4, 6, 12, 12, 20, 30, 60, 72, 144, 216, 336, 336, 672, 864, 1728, 2160, 3200, 4800, 9600, 10560, 14784, 22176, 28224, 35280, 70560, 86400, 172800, 172800, 245760, 368640, 497664, 559872, 1119744, 1679616, 2363904, 2626560, 5253120, 6451200, 12902400, 16128000

Offset: 0

Author

Gus Wiseman, Aug 23 2020

Keywords

• nonn

Examples

			The a(1) = 1 through a(8) = 12 divisors:
  1  1  1  1  1   1   1    1
        3  3  3   3   3    3
              5   5   5    5
              15  9   7    7
                  15  9    9
                  45  15   15
                      21   21
                      35   35
                      45   45
                      63   63
                      105  105
                      315  315
		

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..10000

Crossrefs

A049606 gives the maximum among these divisors, with quotient A060818.

A337257 is the even version.

A000265 gives the maximum odd divisor of n.

A001227 counts odd divisors.

A183063 counts even divisors.

Cf. A000005, A001013, A001055, A006939, A113474, A124010, A253249.

Factorial numbers: A000142, A022559, A027423 (divisors), A048656, A071626, A076716 (factorizations), A325272, A325273, A325617, A336414, A336498.

Programs

Mathematica

Table[Length[Select[Divisors[n!],OddQ]],{n,0,15}]

PARI

a(n) = sumdiv(n!, d, d%2); \\ Michel Marcus, Aug 24 2020

PARI

a(n) = numdiv(prod(k=1, n, k >> valuation(k, 2))); \\ Michel Marcus, Aug 27 2020

Formula

a(n) = A001227(n!).

a(n) = A000005(A049606(n)).

a(n) + A337257(n) = A027423(n) = A000005(n!).

From Seiichi Manyama, Aug 27 2020: (Start)

If p is odd prime, a(p) = 2 * a(p-1).

a(n) = A027423(n) / A113474(n) for n > 0. (End)

Extensions

a(36)-a(44) from Seiichi Manyama, Aug 26 2020