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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


%S 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,

%T 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,

%U 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

%N Least k such that the k-th arithmetic derivative of n is zero, or 0 if no k exists.

%C 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.

%D See A003415

%H T. D. Noe, <a href="/A099307/b099307.txt">Table of n, a(n) for n = 1..10000</a>

%t 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}]

%Y Cf. A003415 (arithmetic derivative of n).

%Y Cf. A099308 (numbers whose k-th arithmetic derivative is zero for some k).

%Y Cf. A099309 (numbers whose k-th arithmetic derivative is nonzero for all k).

%Y Cf. A189760 (least number whose n-th arithmetic derivative is zero).

%K nonn

%O 1,2

%A _T. D. Noe_, Oct 12 2004

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