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

%I A248784 #12 Oct 27 2023 20:46:05
%S A248784 1,1,2,6,10,42,72,360,672,2160,3960,19488,30464,177840,356400,1201200,
%T A248784 2096640,10967040,17510400,121176000,193564800,783455904,1324670976,
%U A248784 8010737280,13121514000,50323046400,88690140000,274271961600,444141105408,2312335872000
%N A248784 Number of squares that divide 1!*2!*3!*...*n!.
%H A248784 Alois P. Heinz, <a href="/A248784/b248784.txt">Table of n, a(n) for n = 1..1000</a> (first 400 terms from Clark Kimberling)
%e A248784 a(4) counts these squares that divide 288:  1, 4, 9, 16, 36, 144.
%p A248784 b:= proc(n) option remember; add(i[2]*x^numtheory[pi](i[1]),
%p A248784       i=ifactors(n)[2])+`if`(n=1, 0, b(n-1))
%p A248784     end:
%p A248784 c:= proc(n) option remember; b(n)+`if`(n=1, 0, c(n-1)) end:
%p A248784 a:= n->(p->mul(iquo(coeff(p, x, i), 2)+1, i=1..degree(p)))(c(n)):
%p A248784 seq(a(n), n=1..30);  # _Alois P. Heinz_, Oct 16 2014
%t A248784 z = 40; p[n_] := Product[k!, {k, 1, n}];
%t A248784 f[n_] := f[n] = FactorInteger[p[n]];
%t A248784 r[m_, x_] := r[m, x] = m*Floor[x/m]
%t A248784 u[n_] := Table[f[n][[i, 1]], {i, 1, Length[f[n]]}];
%t A248784 v[n_] := Table[f[n][[i, 2]], {i, 1, Length[f[n]]}];
%t A248784 t[m_, n_] := Apply[Times, 1 + r[m, v[n]]/m]
%t A248784 m = 2; Table[t[m, n], {n, 1, z}] (* A248784 *)
%Y A248784 Cf. A000178, A000290, A248821, A248822, A248823.
%K A248784 nonn,easy
%O A248784 1,3
%A A248784 _Clark Kimberling_, Oct 15 2014