A068346 a(n) = n'' = second arithmetic derivative of n.

0, 0, 0, 0, 4, 0, 1, 0, 16, 5, 1, 0, 32, 0, 6, 12, 80, 0, 10, 0, 44, 7, 1, 0, 48, 7, 8, 27, 80, 0, 1, 0, 176, 9, 1, 16, 92, 0, 10, 32, 72, 0, 1, 0, 112, 16, 10, 0, 240, 9, 39, 24, 92, 0, 108, 32, 96, 13, 1, 0, 96, 0, 14, 20, 640, 21, 1, 0, 156, 15, 1, 0, 220, 0, 16, 16, 176, 21, 1, 0, 368, 216

Offset: 0

Author

Reinhard Zumkeller, Feb 28 2002

Keywords

• nonn
• look

Comments

a(2p) = 1 for any prime p implies p,p+2 form a twin prime pair. - Kevin J. Gomez, Aug 29 2017

Indices of records > 0 appear to all belong to A116882. - Bill McEachen, Oct 16 2023

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..10000 (first 2000 terms from T. D. Noe)
• Victor Ufnarovski and Bo Åhlander, How to Differentiate a Number, J. Integer Seqs., Vol. 6, 2003.

Crossrefs

Cf. A003415 (arithmetic derivative of n), A099306 (third arithmetic derivative of n).

Column k=2 of A258651.

Programs

Haskell

a068346 = a003415 . a003415  -- Reinhard Zumkeller, Nov 10 2013

Maple

d:= n-> n*add(i[2]/i[1], i=ifactors(n)[2]):
a:= n-> d(d(n));
seq(a(n), n=0..100);  # Alois P. Heinz, Aug 29 2017

Mathematica

dn[0]=0; dn[1]=0; dn[n_]:=Module[{f=Transpose[FactorInteger[n]]}, If[PrimeQ[n], 1, Plus@@(n*f[[2]]/f[[1]])]]; Table[dn[dn[n]], {n, 100}] (T. D. Noe)
f[n_] := If[ Abs@ n < 2, 0, n*Total[#2/#1 & @@@ FactorInteger[Abs@ n]]]; Table[ f[ f[ n]], {n, 81}] (* Robert G. Wilson v, May 12 2012 *)

Formula

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

a(A000040(n)) = 0; a(A157037(n)) = 1. - Reinhard Zumkeller, Feb 22 2009

Extensions

More terms from T. D. Noe, Oct 12 2004