A248783 Number of integers k^6 that divide n!.

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, 36, 36, 36, 36, 36, 36, 36, 48, 56, 56, 112, 112, 112, 112, 128, 128, 128, 128, 192, 192, 216, 216, 270, 270, 270, 270, 300, 300, 300, 300, 300, 360, 396, 396

Offset: 1

Author

Clark Kimberling, Oct 15 2014

Keywords

• nonn
• easy

Examples

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

Links

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

Crossrefs

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

Programs

Mathematica

z = 130; m = 6;
f[n_] := f[n] = FactorInteger[n!]; r[x_] := r[x] = m*Floor[x/m]
u[n_] := Table[f[n][[i, 1]], {i, 1, Length[f[n]]}];
v[n_] := Table[f[n][[i, 2]], {i, 1, Length[f[n]]}];
a[n_] := Apply[Times, 1 + r[v[n]]/m]
t = Table[a[n], {n, 1, z}] (* A248783 *)

PARI

a(n)=c=0;d=divisors(n!);for(i=1,#d,if(ispower(d[i])&&ispower(d[i])%6==0,c++));c+1
n=1;while(n<50,print1(a(n),", ");n++) \\ Derek Orr, Oct 20 2014