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A248780 Number of cubes that divide n!

%I A248780 #15 Mar 27 2015 14:28:50
%S A248780 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 A248780 192,240,270,270,270,270,330,792,792,792,864,864,864,2016,2912,2912,
%U A248780 4704,4704,4704,5376,5760,5760,6144,6144,7680,15360,16320,16320,18360
%N A248780 Number of cubes that divide n!
%H A248780 Clark Kimberling, <a href="/A248780/b248780.txt">Table of n, a(n) for n = 1..1000</a>
%F A248780 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 A248780 a(n) = A061704(A000142(n)). - _Michel Marcus_, Mar 27 2015
%e A248780 a(9) counts these divisors of 9!:  1, 8, 27, 64, 216, 1728.
%t A248780 z = 130; m = 3; f[n_] := f[n] = FactorInteger[n!];
%t A248780 v[n_] := Table[f[n][[i, 2]], {i, 1, Length[f[n]]}];
%t A248780 a[n_] := Apply[Times, 1 + Floor[v[n]/m]]
%t A248780 A248780 = Table[a[n], {n, 1, z}] (* simplified by _M. F. Hasler_, Oct 22 2014 *)
%o A248780 (PARI) a(n)=sumdiv(n!,d,ispower(d,3))
%o A248780 for(n=1,50,print1(a(n),", ")) \\ _Derek Orr_, Oct 20 2014, simplified by _M. F. Hasler_, Oct 22 2014
%o A248780 (PARI) A248780(n)=prod(i=1,#n=factor(n!)[,2],1+n[i]\3) \\ _M. F. Hasler_, Oct 22 2014
%Y A248780 Cf. A000142, A055993, A061704, A248781, A248762.
%K A248780 nonn,easy
%O A248780 1,4
%A A248780 _Clark Kimberling_, Oct 15 2014