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%I #12 Mar 24 2017 13:47:41
%S 0,0,1,7,52,332,2832,22704,242112,2662848,30620160,354965760,
%T 5057925120,68627036160,1054183818240,17469144806400,321351896678400,
%U 5609441772748800,112707637036646400,2192664093342105600,47745925079924736000,1065919878891012096000
%N The pi-based arithmetic derivative of n!.
%H Alois P. Heinz, <a href="/A258845/b258845.txt">Table of n, a(n) for n = 0..450</a>
%F a(n) = A258851(n!) = A258851(A000142(n)).
%p with(numtheory):
%p d:= n-> n*add(i[2]*pi(i[1])/i[1], i=ifactors(n)[2]):
%p a:= proc(n) option remember;
%p `if`(n<2, 0, a(n-1)*n+(n-1)!*d(n))
%p end:
%p seq(a(n), n=0..25);
%t d[n_] := n*Sum[i[[2]]*PrimePi[i[[1]]]/i[[1]], {i, FactorInteger[n]}];
%t a[n_] := a[n] = If[n < 2, 0, a[n-1]*n + (n-1)!*d[n]];
%t Table[a[n], {n, 0, 25}] (* _Jean-François Alcover_, Mar 24 2017, translated from Maple *)
%Y Cf. A000142, A000720, A258851, A259409.
%K nonn
%O 0,4
%A _Alois P. Heinz_, Jun 12 2015
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