A248821 Number of cubes that divide 1!*2!*3!*...*n!.

1, 1, 1, 2, 6, 10, 36, 64, 220, 468, 1024, 2052, 7590, 16224, 50400, 142800, 246240, 510300, 2261952, 3545856, 14152320, 40986000, 68428800, 178293960, 784274400, 1526805504, 2782080000, 9307872000, 15858633600, 28225260000, 143730892800, 225167040000

Offset: 1

Author

Clark Kimberling, Oct 15 2014

Keywords

• nonn
• easy

Examples

			a(5) counts these cubes that divide 34560:  1^3, 2^3, 3^3, 4^3, 6^3, 12^3.
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 1..1000 (first 400 terms from Clark Kimberling)

Crossrefs

Cf. A000178, A000578, A248784, A248822, A248823.

Programs

Maple

b:= proc(n) option remember; add(i[2]*x^numtheory[pi](i[1]),
      i=ifactors(n)[2])+`if`(n=1, 0, b(n-1))
    end:
c:= proc(n) option remember; b(n)+`if`(n=1, 0, c(n-1)) end:
a:= n->(p->mul(iquo(coeff(p, x, i), 3)+1, i=1..degree(p)))(c(n)):
seq(a(n), n=1..30);  # Alois P. Heinz, Oct 16 2014

Mathematica

z = 40; p[n_] := Product[k!, {k, 1, n}];
f[n_] := f[n] = FactorInteger[p[n]];
r[m_, x_] := r[m, 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]]}];
t[m_, n_] := Apply[Times, 1 + r[m, v[n]]/m]
m = 3; Table[t[m, n], {n, 1, z}] (* A248821 *)