A248824 Number of integers k^6 that divide 1!*2!*3!*...*n!.

1, 1, 1, 1, 2, 3, 6, 8, 18, 42, 64, 200, 432, 546, 960, 3888, 6000, 15180, 29952, 38976, 67200, 285600, 393984, 1632960, 3175200, 4165392, 6105600, 38413440, 55339200, 114048000, 205632000, 280219500, 448156800, 2621445120, 3777725952, 12940849152

Offset: 1

Author

Clark Kimberling, Oct 15 2014

Keywords

• nonn
• easy

Examples

			a(7) counts these integers k^6 that divide 125411328000 = A000178(6):  1, 64, 729, 4096, 46656, 2985984, these being k^6 for k = 1, 2, 3, 4, 6, 12.
		

Links

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

Crossrefs

Cf. A000178, A248784, A248821, 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), 6)+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 = 6; Table[t[m, n], {n, 1, z}] (* A248824 *)