A248780 Number of cubes that divide n!

1, 1, 1, 2, 2, 2, 2, 3, 6, 6, 6, 8, 8, 8, 24, 36, 36, 36, 36, 42, 112, 112, 112, 128, 192, 192, 240, 270, 270, 270, 270, 330, 792, 792, 792, 864, 864, 864, 2016, 2912, 2912, 4704, 4704, 4704, 5376, 5760, 5760, 6144, 6144, 7680, 15360, 16320, 16320, 18360

Offset

1,4

Links

Clark Kimberling, Table of n, a(n) for n = 1..1000

Formula

a(n) = product_{i=1..r} 1+floor(e[i]/3), where product_{i=1..r} p[i]^e[i] is the prime factorization of n!. - M. F. Hasler, Oct 22 2014

a(n) = A061704(A000142(n)). - Michel Marcus, Mar 27 2015

Example

a(9) counts these divisors of 9!:  1, 8, 27, 64, 216, 1728.

Mathematica

z = 130; m = 3; f[n_] := f[n] = FactorInteger[n!];
v[n_] := Table[f[n][[i, 2]], {i, 1, Length[f[n]]}];
a[n_] := Apply[Times, 1 + Floor[v[n]/m]]
A248780 = Table[a[n], {n, 1, z}] (* simplified by M. F. Hasler, Oct 22 2014 *)

Prog

(PARI) a(n)=sumdiv(n!, d, ispower(d, 3))
for(n=1, 50, print1(a(n), ", ")) \\ Derek Orr, Oct 20 2014, simplified by M. F. Hasler, Oct 22 2014
(PARI) A248780(n)=prod(i=1, #n=factor(n!)[, 2], 1+n[i]\3) \\ M. F. Hasler, Oct 22 2014

Crossrefs

Cf. A000142, A055993, A061704, A248781, A248762.

Sequence in context: A138705 A333528 A319187 * A213021 A078228 A119762

Adjacent sequences: A248777 A248778 A248779 * A248781 A248782 A248783

Keyword

nonn,easy

Author

Clark Kimberling, Oct 15 2014

Status

approved