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

%I A248823 #11 Oct 27 2023 20:44:53
%S A248823 1,1,1,2,2,6,8,10,42,64,200,432,588,1024,3888,6300,21120,33696,52080,
%T A248823 114240,328320,816480,3326400,4435200,6469632,20616960,57153600,
%U A248823 145411200,258003900,320973840,791513856,1634592960,6403719168,9967104000,34939296000
%N A248823 Number of integers k^5 that divide 1!*2!*3!*...*n!.
%H A248823 Alois P. Heinz, <a href="/A248823/b248823.txt">Table of n, a(n) for n = 1..1000</a> (first 400 terms from Clark Kimberling)
%e A248823 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.
%p A248823 b:= proc(n) option remember; add(i[2]*x^numtheory[pi](i[1]),
%p A248823       i=ifactors(n)[2])+`if`(n=1, 0, b(n-1))
%p A248823     end:
%p A248823 c:= proc(n) option remember; b(n)+`if`(n=1, 0, c(n-1)) end:
%p A248823 a:= n->(p->mul(iquo(coeff(p, x, i), 5)+1, i=1..degree(p)))(c(n)):
%p A248823 seq(a(n), n=1..30);  # _Alois P. Heinz_, Oct 16 2014
%t A248823 z = 40; p[n_] := Product[k!, {k, 1, n}];
%t A248823 f[n_] := f[n] = FactorInteger[p[n]];
%t A248823 r[m_, x_] := r[m, x] = m*Floor[x/m]
%t A248823 u[n_] := Table[f[n][[i, 1]], {i, 1, Length[f[n]]}];
%t A248823 v[n_] := Table[f[n][[i, 2]], {i, 1, Length[f[n]]}];
%t A248823 t[m_, n_] := Apply[Times, 1 + r[m, v[n]]/m]
%t A248823 m = 5; Table[t[m, n], {n, 1, z}] (* A248823 *)
%Y A248823 Cf. A000178, A248784, A248821, A248822.
%K A248823 nonn,easy
%O A248823 1,4
%A A248823 _Clark Kimberling_, Oct 15 2014