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A099307
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Least k such that the k-th arithmetic derivative of n is zero, or 0 if no k exists.
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6
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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; internal format)
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OFFSET
| 1,2
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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.
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REFERENCES
| See A003415
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LINKS
| T. D. Noe, Table of n, a(n) for n = 1..10000
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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}]
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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: A141661 A080769 A143539 * A072738 A165316 A141058
Adjacent sequences: A099304 A099305 A099306 * A099308 A099309 A099310
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KEYWORD
| nonn
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AUTHOR
| T. D. Noe (noe(AT)sspectra.com), Oct 12 2004
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