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%I #32 Jan 05 2023 16:15:36
%S -1,-1,0,-1,-2,-1,2,-3,-2,-1,9,-1,-4,23,1,-1,-4,-1,5,-2,-2,-1,2,-3,24,
%T 0,18,-1,-2,-1,6,-5,-2,85,7,-1,-4,21,10,-1,-2,-1,35,53,-4,-1,2,-5,44,
%U 18,34,-1,2,21,4,-3,-2,-1,16,-1,-6,21,1,-5,-2,-1,7,85,-2,-1,4,-1,23,55,5,-4,-2,-1,4,9,-2,-1,42,-3,42
%N a(n) gives the number of iterations of x -> A003415(x) needed to reach the first number which is either a divisor or multiple of n, but not both at the same time. If no such number can ever be reached, a(n) is 0 (when either n is of the form p^p, or if the iteration would never stop). When the number reached is a divisor of n, a(n) is -1 * iteration count.
%C The absolute value of a(n) tells how many columns right from the leftmost column in array A258651 one needs to go at row n, before one (again) finds either a divisor or a multiple of n, with 0's reserved for cases like 4 and 27 where the same value continues forever. If one finds a divisor before a multiple, then the value of a(n) will be negative, otherwise it will be positive.
%C Question: What is the value of a(91) ?
%H Antti Karttunen, <a href="/A328383/b328383.txt">Table of n, a(n) for n = 2..90</a>
%F a(A000040(n)) = -1.
%F a(A051674(n)) = 0.
%e For n = 6, its arithmetic derivative A003415(6) = 5 is neither its divisor nor its multiple, but the second arithmetic derivative A003415(5) = 1 is its divisor, thus a(6) = -2.
%e For n = 8, its arithmetic derivative A003415(8) = 12 is neither its divisor nor its multiple, but the second arithmetic derivative A003415(12) = 16 is its multiple, thus a(8) = +2.
%e Numbers reached for n=2..28 (with positions of the form p^p are filled with the same p^p): 1, 1, 4, 1, 1, 1, 16, 1, 1, 1, 8592, 1, 1, 410267320320, 32, 1, 1, 1, 240, 7, 1, 1, 48, 1, 410267320320, 27, 9541095424. For example, we have a(12) = 9 and the 9th arithmetic derivative of 12 is A003415^(9)(12) = 8592 = 716*12.
%o (PARI)
%o A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
%o A328383(n) = { my(u=A003415(n),k=1); if(u==n,return(0)); while((n%u) && (u%n), k++; u = A003415(u)); if(u%n,-k,k); };
%Y Cf. A003415, A258651, A328235, A328236.
%Y Cf. A051674 (indices of zeros provided for all n >= 2 either a divisor or multiple can be found).
%Y Cf. A256750, A328248, A328384 for similar counts.
%K sign,hard,more
%O 2,5
%A _Antti Karttunen_, Oct 15 2019
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