A248781 Number of integers k^4 that divide n!

1, 1, 1, 1, 1, 2, 2, 2, 4, 6, 6, 6, 6, 6, 6, 8, 8, 15, 15, 30, 30, 30, 30, 36, 36, 36, 48, 112, 112, 112, 112, 128, 128, 144, 216, 270, 270, 270, 270, 300, 300, 300, 300, 660, 792, 792, 792, 864, 1296, 1728, 1728, 3744, 3744, 4368, 4368, 4704, 4704, 4704

Offset: 1

Author

Clark Kimberling, Oct 15 2014

Keywords

• nonn
• easy

Examples

			a(10) counts these divisors of 10!:  1, 16, 81, 256, 1296, 20736.
		

Links

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

Crossrefs

Cf. A000142, A055993, A248780, A248765.

Programs

Mathematica

z = 130; m = 4;
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}] (* A248781 *)

PARI

a(n)=c=0;d=divisors(n!);for(i=1,#d,if(ispower(d[i])&&ispower(d[i])%4==0,c++));c+1
n=1;while(n<50,print1(a(n),", ");n++) \\ Derek Orr, Oct 20 2014