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URL: https://oeis.org/A351089

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A351089
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.
7
-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, -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, 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
OFFSET
0,9
EXAMPLE
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.
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.
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.
a(6) = 0 because 6 is already >= A276086(6) = 5 before any iterations.
a(8) = 2 because it takes two iterations with A003415 as 8 -> 12 -> 16 to obtain a number >= A276086(8) = 15.
PROG
(PARI)
A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
A276086(n) = { my(m=1, p=2); while(n, m *= (p^(n%p)); n = n\p; p = nextprime(1+p)); (m); };
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); };
CROSSREFS
Cf. A003415, A276086, A349908 (positions of records), A351226 (positions of zeros), A351229 (positions of ones).
Cf. also A351088.
Sequence in context: A290084 A154844 A383153 * A133831 A325613 A305054
KEYWORD
sign,base,look
AUTHOR
Antti Karttunen, Feb 05 2022
STATUS
approved