A248784 Number of squares that divide 1!*2!*3!*...*n!.

1, 1, 2, 6, 10, 42, 72, 360, 672, 2160, 3960, 19488, 30464, 177840, 356400, 1201200, 2096640, 10967040, 17510400, 121176000, 193564800, 783455904, 1324670976, 8010737280, 13121514000, 50323046400, 88690140000, 274271961600, 444141105408, 2312335872000

Offset: 1

Author

Clark Kimberling, Oct 15 2014

Keywords

• nonn
• easy

Examples

			a(4) counts these squares that divide 288:  1, 4, 9, 16, 36, 144.
		

Links

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

Crossrefs

Cf. A000178, A000290, 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), 2)+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 = 2; Table[t[m, n], {n, 1, z}] (* A248784 *)