A258845 The pi-based arithmetic derivative of n!.

0, 0, 1, 7, 52, 332, 2832, 22704, 242112, 2662848, 30620160, 354965760, 5057925120, 68627036160, 1054183818240, 17469144806400, 321351896678400, 5609441772748800, 112707637036646400, 2192664093342105600, 47745925079924736000, 1065919878891012096000

Offset

0,4

Links

Alois P. Heinz, Table of n, a(n) for n = 0..450

Formula

a(n) = A258851(n!) = A258851(A000142(n)).

Maple

with(numtheory):
d:= n-> n*add(i[2]*pi(i[1])/i[1], i=ifactors(n)[2]):
a:= proc(n) option remember;
      `if`(n<2, 0, a(n-1)*n+(n-1)!*d(n))
    end:
seq(a(n), n=0..25);

Mathematica

d[n_] := n*Sum[i[[2]]*PrimePi[i[[1]]]/i[[1]], {i, FactorInteger[n]}];
a[n_] := a[n] = If[n < 2, 0, a[n-1]*n + (n-1)!*d[n]];
Table[a[n], {n, 0, 25}] (* Jean-François Alcover, Mar 24 2017, translated from Maple *)

Crossrefs

Cf. A000142, A000720, A258851, A259409.

Sequence in context: A206809 A027542 A254946 * A037593 A037684 A246513

Adjacent sequences: A258842 A258843 A258844 * A258846 A258847 A258848

Keyword

nonn

Author

Alois P. Heinz, Jun 12 2015

Status

approved