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

0,5

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.

Formula

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

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

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 *)

Prog

(Haskell)
a068346 = a003415 . a003415  -- Reinhard Zumkeller, Nov 10 2013

Crossrefs

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

Column k=2 of A258651.

Sequence in context: A136448 A166318 A166317 * A348304 A006838 A061309

Adjacent sequences: A068343 A068344 A068345 * A068347 A068348 A068349

Keyword

nonn,look

Author

Reinhard Zumkeller, Feb 28 2002

Extensions

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

Status

approved