A248821 - OEIS

%I #11 Oct 27 2023 20:45:41

%S 1,1,1,2,6,10,36,64,220,468,1024,2052,7590,16224,50400,142800,246240,

%T 510300,2261952,3545856,14152320,40986000,68428800,178293960,

%U 784274400,1526805504,2782080000,9307872000,15858633600,28225260000,143730892800,225167040000

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

%H Alois P. Heinz, <a href="/A248821/b248821.txt">Table of n, a(n) for n = 1..1000</a> (first 400 terms from Clark Kimberling)

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

%p b:= proc(n) option remember; add(i[2]*x^numtheory[pi](i[1]),

%p       i=ifactors(n)[2])+`if`(n=1, 0, b(n-1))

%p     end:

%p c:= proc(n) option remember; b(n)+`if`(n=1, 0, c(n-1)) end:

%p a:= n->(p->mul(iquo(coeff(p, x, i), 3)+1, i=1..degree(p)))(c(n)):

%p seq(a(n), n=1..30);  # _Alois P. Heinz_, Oct 16 2014

%t z = 40; p[n_] := Product[k!, {k, 1, n}];

%t f[n_] := f[n] = FactorInteger[p[n]];

%t r[m_, x_] := r[m, x] = m*Floor[x/m]

%t u[n_] := Table[f[n][[i, 1]], {i, 1, Length[f[n]]}];

%t v[n_] := Table[f[n][[i, 2]], {i, 1, Length[f[n]]}];

%t t[m_, n_] := Apply[Times, 1 + r[m, v[n]]/m]

%t m = 3; Table[t[m, n], {n, 1, z}] (* A248821 *)

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

%K nonn,easy

%O 1,4

%A _Clark Kimberling_, Oct 15 2014