%I #15 Mar 27 2015 14:28:50

%S 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,

%T 192,240,270,270,270,270,330,792,792,792,864,864,864,2016,2912,2912,

%U 4704,4704,4704,5376,5760,5760,6144,6144,7680,15360,16320,16320,18360

%N Number of cubes that divide n!

%H Clark Kimberling, <a href="/A248780/b248780.txt">Table of n, a(n) for n = 1..1000</a>

%F 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

%F a(n) = A061704(A000142(n)). - _Michel Marcus_, Mar 27 2015

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

%t z = 130; m = 3; f[n_] := f[n] = FactorInteger[n!];

%t v[n_] := Table[f[n][[i, 2]], {i, 1, Length[f[n]]}];

%t a[n_] := Apply[Times, 1 + Floor[v[n]/m]]

%t A248780 = Table[a[n], {n, 1, z}] (* simplified by _M. F. Hasler_, Oct 22 2014 *)

%o (PARI) a(n)=sumdiv(n!,d,ispower(d,3))

%o for(n=1,50,print1(a(n),", ")) \\ _Derek Orr_, Oct 20 2014, simplified by _M. F. Hasler_, Oct 22 2014

%o (PARI) A248780(n)=prod(i=1,#n=factor(n!)[,2],1+n[i]\3) \\ _M. F. Hasler_, Oct 22 2014

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

%K nonn,easy

%O 1,4

%A _Clark Kimberling_, Oct 15 2014