A248822 Number of integers k^4 that divide 1!*2!*3!*...*n!.

1, 1, 1, 2, 3, 8, 10, 36, 64, 200, 432, 630, 1088, 4800, 7590, 32448, 47040, 114240, 164160, 835920, 1302840, 4804800, 7091712, 25243920, 39168000, 171555840, 320973840, 667447200, 1113944832, 3338108928, 5181926400, 19372953600, 31804416000, 132562944000

Offset: 1

Author

Clark Kimberling, Oct 15 2014

Keywords

• nonn
• easy

Examples

			a(6) counts these integers k^4 that divide 24883200:  1^4, 2^4, 4^4, 8^4, 6^4, 12^4, 24^4.
		

Links

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

Crossrefs

Cf. A000178, A056571, A248784, A248821, 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), 4)+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 = 4; Table[t[m, n], {n, 1, z}] (* A248822 *)