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

%I A248824 #10 Oct 27 2023 19:16:26
%S A248824 1,1,1,1,2,3,6,8,18,42,64,200,432,546,960,3888,6000,15180,29952,38976,
%T A248824 67200,285600,393984,1632960,3175200,4165392,6105600,38413440,
%U A248824 55339200,114048000,205632000,280219500,448156800,2621445120,3777725952,12940849152
%N A248824 Number of integers k^6 that divide 1!*2!*3!*...*n!.
%H A248824 Alois P. Heinz, <a href="/A248824/b248824.txt">Table of n, a(n) for n = 1..1000</a> (first 400 terms from Clark Kimberling)
%e A248824 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.
%p A248824 b:= proc(n) option remember; add(i[2]*x^numtheory[pi](i[1]),
%p A248824       i=ifactors(n)[2])+`if`(n=1, 0, b(n-1))
%p A248824     end:
%p A248824 c:= proc(n) option remember; b(n)+`if`(n=1, 0, c(n-1)) end:
%p A248824 a:= n->(p->mul(iquo(coeff(p, x, i), 6)+1, i=1..degree(p)))(c(n)):
%p A248824 seq(a(n), n=1..30);  # _Alois P. Heinz_, Oct 16 2014
%t A248824 z = 40; p[n_] := Product[k!, {k, 1, n}];
%t A248824 f[n_] := f[n] = FactorInteger[p[n]];
%t A248824 r[m_, x_] := r[m, x] = m*Floor[x/m]
%t A248824 u[n_] := Table[f[n][[i, 1]], {i, 1, Length[f[n]]}];
%t A248824 v[n_] := Table[f[n][[i, 2]], {i, 1, Length[f[n]]}];
%t A248824 t[m_, n_] := Apply[Times, 1 + r[m, v[n]]/m]
%t A248824 m = 6; Table[t[m, n], {n, 1, z}] (* A248824 *)
%Y A248824 Cf. A000178, A248784, A248821, A248822, A248823.
%K A248824 nonn,easy
%O A248824 1,5
%A A248824 _Clark Kimberling_, Oct 15 2014