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Let b_k=3...3 consist of k>=1 3's. Then a(n) is the smallest k such that the concatenation 2^n b_k is prime, or a(n)=0 if there is no such prime.
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%I #9 Sep 17 2014 15:54:13

%S 1,1,1,1,1,4,1,1,2,3,1,1,4,4,6,30,3,1,6,1,32,3,3,2,22,1,6,1,2,14,7,1,

%T 10,1,2,6,3,4,2,5,2,6,1,1,37,53,53,13,64,1,67,1,45,29,17,12,14,1,2,5,

%U 15,36,10,7,1,1,81,4,18,5,55,8,33,19,8,6,2,11

%N Let b_k=3...3 consist of k>=1 3's. Then a(n) is the smallest k such that the concatenation 2^n b_k is prime, or a(n)=0 if there is no such prime.

%C Conjecture: for all n, a(n)>0.

%o (PARI) a(n) = {k = 0; while (! isprime(eval(concat(Str(2^n), Str((10^k-1)/3)))), k++); k;} \\ _Michel Marcus_, Sep 16 2014

%Y Cf. A000079, A232210, A242775.

%K nonn

%O 0,6

%A _Vladimir Shevelev_, Sep 14 2014

%E More terms from _Michel Marcus_, Sep 16 2014