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

0,4

Links

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

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)

Example

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

Mathematica

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

Prog

(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

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.

Sequence in context: A318847 A228892 A267610 * A291365 A154779 A332983

Adjacent sequences: A336937 A336938 A336939 * A336941 A336942 A336943

Keyword

nonn

Author

Gus Wiseman, Aug 23 2020

Extensions

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

Status

approved