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

Author

Clark Kimberling, Oct 15 2014

Keywords

• nonn
• easy

Examples

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

Links

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

Crossrefs

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

Programs

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 *)

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

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