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 A068346 a(n) = n'' = second arithmetic derivative of n. 24
 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 April 11 00:03 EDT 2021. Contains 342877 sequences. (Running on oeis4.)