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A068346 a(n) = n'' = second arithmetic derivative of n. 22
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,5

COMMENTS

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

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

LINKS

T. D. Noe and Alois P. Heinz, Table of n, a(n) for n = 0..10000 (first 2000 terms from T. D. Noe)

Victor Ufnarovski and Bo Ã…hlander, How to Differentiate a Number, J. Integer Seqs., Vol. 6, 2003.

FORMULA

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

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 * A006838 A061309 A263655

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

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Last modified November 15 01:25 EST 2019. Contains 329143 sequences. (Running on oeis4.)