A192016 Second arithmetic derivative of prime powers: a(n) = A068346(A000961(n)).

0, 0, 0, 4, 0, 0, 16, 5, 0, 0, 80, 0, 0, 0, 7, 27, 0, 0, 176, 0, 0, 0, 0, 9, 0, 0, 0, 640, 0, 0, 0, 0, 216, 0, 0, 0, 0, 0, 0, 0, 0, 13, 55, 0, 1408, 0, 0, 0, 0, 0, 0, 0, 0, 15, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 621, 0, 5120, 0, 0, 0, 0, 0, 0, 0, 19

Offset: 1

Author

Reinhard Zumkeller, Jun 26 2011

Keywords

• nonn

Links

• Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A000040. A000961, A002808, A003415, A068346, A192015.

Programs

Haskell

a192016 = a068346 . a000961  -- Reinhard Zumkeller, Apr 16 2014

Mathematica

ad[n_] := n * Plus @@ ((Last[#]/First[#]) & /@ FactorInteger[n]); ad[0] = ad[1] = 0; f[n_] := If[n == 1, 0, If[PrimePowerQ[n], {p, e} = FactorInteger[n][[1]]; e*p^(e-1), Nothing]]; ad /@ Array[f, 300] (* Amiram Eldar, Apr 11 2025 *)

Formula

a(n) = A003415(A192015(n)).

a(A000040(n)) = 0; a(A002808(n)) > 0.