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 A099307 Least k such that the k-th arithmetic derivative of n is zero, or 0 if no k exists. 9
 1, 2, 2, 0, 2, 3, 2, 0, 4, 3, 2, 0, 2, 5, 0, 0, 2, 5, 2, 0, 4, 3, 2, 0, 4, 0, 0, 0, 2, 3, 2, 0, 6, 3, 0, 0, 2, 5, 0, 0, 2, 3, 2, 0, 0, 5, 2, 0, 6, 0, 0, 0, 2, 0, 0, 0, 4, 3, 2, 0, 2, 7, 0, 0, 6, 3, 2, 0, 0, 3, 2, 0, 2, 0, 0, 0, 6, 3, 2, 0, 0, 3, 2, 0, 4, 0, 0, 0, 2, 0, 0, 0, 4, 7, 0, 0, 2, 7, 0, 0, 2, 0, 2, 0, 3 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 COMMENTS Denote the k-th derivative of n by d(n,k). We know that we can stop taking derivatives if either d(n,k) = 0 or d(n,k) has a factor of the form p^p for prime p. In the latter case, the derivatives will stay constant or grow without bound. REFERENCES See A003415 LINKS T. D. Noe, Table of n, a(n) for n = 1..10000 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[k=0; d=n; done=False; While[If[d==1, done=True, f=FactorInteger[d]; Do[If[f[[i, 1]]<=f[[i, 2]], done=True], {i, Length[f]}]]; !done, k++; d=dn[d]]; If[d==1, k+1, 0], {n, 200}] CROSSREFS Cf. A003415 (arithmetic derivative of n). Cf. A099308 (numbers whose k-th arithmetic derivative is zero for some k). Cf. A099309 (numbers whose k-th arithmetic derivative is nonzero for all k). Cf. A189760 (least number whose n-th arithmetic derivative is zero). Sequence in context: A143539 A249063 A341879 * A256750 A228430 A241533 Adjacent sequences:  A099304 A099305 A099306 * A099308 A099309 A099310 KEYWORD nonn AUTHOR T. D. Noe, Oct 12 2004 STATUS approved

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Last modified May 16 21:28 EDT 2021. Contains 343951 sequences. (Running on oeis4.)