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%I #12 Sep 15 2022 11:54:11
%S 2,4,1,0,3,1,0,1,1,0,2,0,0,2,1,0,0,1,0,3,1,0,1,0,0,2,0,0,1,1,0,0,1,0,
%T 1,1,0,0,1,0,2,0,0,6,0,0,0,1,0,1,1,0,1,1,0,2,0,0,0,0,0,0,1,0,2,0,0,1,
%U 1,0,0,0,0,1,1,0,0,1,0,0,1,0,1,0,0,2,0,0,5,1
%N a(n) = max{k: f(n),...,f^k(n) are prime}, where f(m) = 2m+1 and f^k denotes composition of f with itself k times.
%C From _Glen Whitney_, Sep 14 2022: (Start)
%C If a(n) > 3 and n > 5, then the final digit of n is 4 or 9.
%C a(n) > 0 if and only if n appears in A005097.
%C More generally, a(n) > m if and only if all of 2^k(n+1) - 1 for 0 <= k <= m are in A005097.
%C Creating a tile labeled by a multiple of p for a prime p with a relatively large value of a(p) is considered valuable in the game DIVE (see links). (End)
%H Glen Whitney, <a href="/A067849/b067849.txt">Table of n, a(n) for n = 1..10000</a>
%H Number-theoretic puzzle game <a href="https://alexfink.github.io/dive/">DIVE</a>
%H Glen Whitney, <a href="/A067849/a067849.py.txt">Python program to compute a(n)</a>
%e f(2) = 5, f(f(2)) = 11, f(f(f(2))) = 23, f(f(f(f(2)))) = 47, all prime, but f^5(2) = 95 is not prime, so a(2) = 4.
%t f[n_] := Module[{a = 2n + 1, i = 0}, While[PrimeQ[a], i++; a = 2a + 1]; i]; Table[f[i], {i, 1, 60}]
%o (PARI) a(n) = {my(nb = 0, newn); while (isprime(newn=2*n+1), nb++; n = newn); nb;} \\ _Michel Marcus_, Nov 10 2018
%K nonn
%O 1,1
%A _Joseph L. Pe_, Feb 14 2002
%E More terms from _Michel Marcus_, Nov 10 2018
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