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.
-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
Examples
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.
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Crossrefs
Programs
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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); };
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