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A336940
Number of odd divisors of n!.
2
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
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.
Factorial numbers: A000142, A022559, A027423 (divisors), A048656, A071626, A076716 (factorizations), A325272, A325273, A325617, A336414, A336498.
Sequence in context: A318847 A228892 A267610 * A291365 A154779 A332983
KEYWORD
nonn
AUTHOR
Gus Wiseman, Aug 23 2020
EXTENSIONS
a(36)-a(44) from Seiichi Manyama, Aug 26 2020
STATUS
approved