A248823 Number of integers k^5 that divide 1!*2!*3!*...*n!.

1, 1, 1, 2, 2, 6, 8, 10, 42, 64, 200, 432, 588, 1024, 3888, 6300, 21120, 33696, 52080, 114240, 328320, 816480, 3326400, 4435200, 6469632, 20616960, 57153600, 145411200, 258003900, 320973840, 791513856, 1634592960, 6403719168, 9967104000, 34939296000

Offset

1,4

Links

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

Example

a(6) counts these integers k^5 that divide 24883200:  1, 32, 1024, 7776, 32768, 248832, these being k^5 for k = 1, 2, 3, 4, 6, 12.

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), 5)+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 = 5; Table[t[m, n], {n, 1, z}] (* A248823 *)

Crossrefs

Cf. A000178, A248784, A248821, A248822.

Sequence in context: A320247 A320248 A320067 * A284616 A136513 A365662

Adjacent sequences: A248820 A248821 A248822 * A248824 A248825 A248826

Keyword

nonn,easy

Author

Clark Kimberling, Oct 15 2014

Status

approved