A248783 - OEIS

%I #8 Oct 27 2014 20:48:28

%S 1,1,1,1,1,1,1,2,2,2,2,2,2,2,4,6,6,6,6,8,8,8,8,8,16,16,24,30,30,30,30,

%T 36,36,36,36,36,36,36,48,56,56,112,112,112,112,128,128,128,128,192,

%U 192,216,216,270,270,270,270,300,300,300,300,300,360,396,396

%N Number of integers k^6 that divide n!.

%H Clark Kimberling, <a href="/A248783/b248783.txt">Table of n, a(n) for n = 1..1000</a>

%e a(16) counts these divisors of 16!:  1^6, 2^6, 2^12, 3^6, 6^6, 12^6.

%t z = 130; m = 6;

%t f[n_] := f[n] = FactorInteger[n!]; r[x_] := r[x] = m*Floor[x/m]

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

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

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

%t t = Table[a[n], {n, 1, z}] (* A248783 *)

%o (PARI) a(n)=c=0;d=divisors(n!);for(i=1,#d,if(ispower(d[i])&&ispower(d[i])%6==0,c++));c+1

%o n=1;while(n<50,print1(a(n),", ");n++) \\ _Derek Orr_, Oct 20 2014

%Y Cf. A000142, A055993, A248780, A248781, A248782, A248771.

%K nonn,easy

%O 1,8

%A _Clark Kimberling_, Oct 15 2014