

A099307


Least k such that the kth 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
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OFFSET

1,2


COMMENTS

Denote the kth 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 kth arithmetic derivative is zero for some k).
Cf. A099309 (numbers whose kth arithmetic derivative is nonzero for all k).
Cf. A189760 (least number whose nth 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



