%I #53 Oct 27 2023 19:47:10

%S 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,

%T 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,

%U 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

%N a(n) = n'' = second arithmetic derivative of n.

%C a(2p) = 1 for any prime p implies p,p+2 form a twin prime pair. - _Kevin J. Gomez_, Aug 29 2017

%C Indices of records > 0 appear to all belong to A116882. - _Bill McEachen_, Oct 16 2023

%H Alois P. Heinz, <a href="/A068346/b068346.txt">Table of n, a(n) for n = 0..10000</a> (first 2000 terms from T. D. Noe)

%H Victor Ufnarovski and Bo Åhlander, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL6/Ufnarovski/ufnarovski.html">How to Differentiate a Number</a>, J. Integer Seqs., Vol. 6, 2003.

%F a(n) = A003415(A003415(n)).

%F a(A000040(n)) = 0; a(A157037(n)) = 1. - _Reinhard Zumkeller_, Feb 22 2009

%p d:= n-> n*add(i[2]/i[1], i=ifactors(n)[2]):

%p a:= n-> d(d(n));

%p seq(a(n), n=0..100); # _Alois P. Heinz_, Aug 29 2017

%t 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)

%t 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 *)

%o (Haskell)

%o a068346 = a003415 . a003415 -- _Reinhard Zumkeller_, Nov 10 2013

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

%Y Column k=2 of A258651.

%K nonn,look

%O 0,5

%A _Reinhard Zumkeller_, Feb 28 2002

%E More terms from _T. D. Noe_, Oct 12 2004