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A327966 Number of iterations of "tamed variant of arithmetic derivative", A327965 needed to reach 0 from n, or -1 if zero is never reached.

%I A327966 #10 Oct 01 2019 19:51:54
%S A327966 0,1,2,2,2,2,3,2,3,4,3,2,2,2,5,3,3,2,5,2,4,4,3,2,3,4,4,2,3,2,3,2,3,6,
%T A327966 3,3,4,2,5,2,3,2,3,2,3,3,5,2,3,6,4,3,6,2,3,2,3,4,3,2,3,2,7,4,3,6,3,2,
%U A327966 6,5,3,2,3,2,3,3,3,6,3,2,3,2,3,2,3,4,4,3,4,2,4,3,4,4,7,4,3,2,7,4,4,2,4,2,3,3,3,2,3,2,4,4,4,2,3,3,4,4,3,4,3
%N A327966 Number of iterations of "tamed variant of arithmetic derivative", A327965 needed to reach 0 from n, or -1 if zero is never reached.
%C A327966 Conjecture: from all n, zero is eventually reached.
%H A327966 Antti Karttunen, <a href="/A327966/b327966.txt">Table of n, a(n) for n = 0..90609</a>
%F A327966 a(0) = 0; for n > 0, a(n) = 1 + a(A327965(n)).
%F A327966 a(p) = 2 for all primes p.
%o A327966 (PARI)
%o A327966 A003415(n) = {my(fac); if(n<1, 0, fac=factor(n); sum(i=1, matsize(fac)[1], n*fac[i, 2]/fac[i, 1]))}; \\ From A003415
%o A327966 A327938(n) = { my(f = factor(n)); for(k=1, #f~, f[k,2] = (f[k,2]%f[k,1])); factorback(f); };
%o A327966 A327965(n) = if(n<=1,0,A327938(A003415(n)));
%o A327966 A327966(n) = { my(k=0); while(n>0, k++; n = A327965(n)); (k); };
%o A327966 \\ Or alternatively, as a recurrence:
%o A327966 A327966(n) = if(!n,0,1+A327966(A327965(n)));
%Y A327966 Cf. A003415, A256750, A327938, A327965, A327967 (indices of the records).
%K A327966 nonn
%O A327966 0,3
%A A327966 _Antti Karttunen_, Oct 01 2019