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%I #16 Feb 05 2022 17:05:27
%S -1,-1,-1,-1,-1,-1,0,-1,2,-1,-1,-1,2,-1,-1,6,4,-1,-1,-1,6,-1,-1,-1,6,
%T -1,11,-1,8,-1,0,0,0,-1,-1,5,0,-1,-1,5,7,-1,-1,-1,4,8,-1,-1,4,-1,10,
%U 10,8,-1,7,10,8,-1,-1,-1,0,-1,-1,8,2,-1,-1,-1,6,11,-1,-1,6,-1,10,10,8,-1,-1,-1,7,9,-1,-1,7,-1,14,11,9
%N Number of iterations of map x -> A003415(x) needed to reach a number >= A276086(n), when starting from x = n, or -1 if such number is never reached.
%H Antti Karttunen, <a href="/A351089/b351089.txt">Table of n, a(n) for n = 0..65537</a>
%H <a href="/index/Pri#primorialbase">Index entries for sequences related to primorial base</a>
%e a(0) = -1 because A003415^(k)(0) = 0 for all values of k >= 0 (i.e., regardless of how many times we apply the arithmetic derivative), and 0 < A276086(0) = 1.
%e a(1) = -1 because A003415^(k)(1) = 0 for all values of k >= 1, and both 1 and 0 are less than A276086(1) = 2.
%e a(4) = -1 because A003415^(k)(4) = 4 for all values of k >= 0 (i.e., regardless of how many times we apply the arithmetic derivative), and 4 < A276086(4) = 9.
%e a(6) = 0 because 6 is already >= A276086(6) = 5 before any iterations.
%e a(8) = 2 because it takes two iterations with A003415 as 8 -> 12 -> 16 to obtain a number >= A276086(8) = 15.
%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 A276086(n) = { my(m=1, p=2); while(n, m *= (p^(n%p)); n = n\p; p = nextprime(1+p)); (m); };
%o A351089(n) = { my(u=A276086(n),i=0,prev_n=-1); while(n>0, if(n>=u, return(i)); prev_n = n; n = A003415(n); if(n==prev_n, return(-1)); i++); (-1); };
%Y Cf. A003415, A276086, A349908 (positions of records), A351226 (positions of zeros), A351229 (positions of ones).
%Y Cf. also A351088.
%K sign,base,look
%O 0,9
%A _Antti Karttunen_, Feb 05 2022