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A248822 Number of integers k^4 that divide 1!*2!*3!*...*n!.

%I A248822 #10 Oct 27 2023 20:45:14
%S A248822 1,1,1,2,3,8,10,36,64,200,432,630,1088,4800,7590,32448,47040,114240,
%T A248822 164160,835920,1302840,4804800,7091712,25243920,39168000,171555840,
%U A248822 320973840,667447200,1113944832,3338108928,5181926400,19372953600,31804416000,132562944000
%N A248822 Number of integers k^4 that divide 1!*2!*3!*...*n!.
%H A248822 Alois P. Heinz, <a href="/A248822/b248822.txt">Table of n, a(n) for n = 1..1000</a> (first 400 terms from Clark Kimberling)
%e A248822 a(6) counts these integers k^4 that divide 24883200:  1^4, 2^4, 4^4, 8^4, 6^4, 12^4, 24^4.
%p A248822 b:= proc(n) option remember; add(i[2]*x^numtheory[pi](i[1]),
%p A248822       i=ifactors(n)[2])+`if`(n=1, 0, b(n-1))
%p A248822     end:
%p A248822 c:= proc(n) option remember; b(n)+`if`(n=1, 0, c(n-1)) end:
%p A248822 a:= n->(p->mul(iquo(coeff(p, x, i), 4)+1, i=1..degree(p)))(c(n)):
%p A248822 seq(a(n), n=1..30);  # _Alois P. Heinz_, Oct 16 2014
%t A248822 z = 40; p[n_] := Product[k!, {k, 1, n}];
%t A248822 f[n_] := f[n] = FactorInteger[p[n]];
%t A248822 r[m_, x_] := r[m, x] = m*Floor[x/m]
%t A248822 u[n_] := Table[f[n][[i, 1]], {i, 1, Length[f[n]]}];
%t A248822 v[n_] := Table[f[n][[i, 2]], {i, 1, Length[f[n]]}];
%t A248822 t[m_, n_] := Apply[Times, 1 + r[m, v[n]]/m]
%t A248822 m = 4; Table[t[m, n], {n, 1, z}] (* A248822 *)
%Y A248822 Cf. A000178, A056571, A248784, A248821, A248823.
%K A248822 nonn,easy
%O A248822 1,4
%A A248822 _Clark Kimberling_, Oct 15 2014