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%I #26 Feb 27 2018 11:13:24
%S 1,1,1,1,1,3,1,1,7,5,3,3,1,5,5,1,1,3,1,7,7,25,13,39,5,7,15,13,7,3,1,5,
%T 9,3,25,3,15,3,5,27,3,9,3,15,7,19,27,5,19,7,17,7,51,5,3,27,29,77,27,
%U 17,1,53,9,3,9,3,9,31,23,27,39,5,15,21,5,3,29
%N a(n) = smallest odd k such that either k*2^n - 1 or k*2^n + 1 is prime.
%C As N increases, (Sum_{n=1..N} a(n)) / (Sum_{n=1..N} n) tends to log(2)/2 as seen by plotting data; this is consistent with the prime number theorem as the probability that x*2^n - 1 and x*2^n + 1 are prime is ~ 2/(n*log(2)) if n is great enough, so after n*log(2)/2 try (n*log(2)/2)*(2/n*log(2))=1.
%C For n=1 to 10000, a(n)/n is always < 3.2.
%C a(n)*2^n - 1 and a(n)*2^n + 1 are twin primes for n = 2, 6, 18, 63, 211, 546, 1032, 1156, 1553, 4901, 8335, 8529; corresponding values of a(n) are 1, 3, 3, 9, 9,297, 177, 1035, 291, 2565, 3975, 459.
%H Pierre CAMI, <a href="/A284631/b284631.txt">Table of n, a(n) for n = 1..10000</a>
%H Pierre CAMI, <a href="/A284631/a284631.txt">PFGW Script</a>
%e 1*2^1 + 1 = 3 (prime), so a(1) = 1;
%e 1*2^2 - 1 = 3 (prime), so a(2) = 1;
%e 1*2^3 - 1 = 7 (prime), so a(3) = 1.
%t Table[k = 1; While[Nor @@ Map[PrimeQ, k*2^n + {-1, 1}], k += 2]; k, {n, 77}] (* _Michael De Vlieger_, Apr 02 2017 *)
%o (PARI) a(n) = my(k=1); while (!isprime(k*2^n-1) && !isprime(k*2^n+1), k+=2); k; \\ _Michel Marcus_, Mar 31 2017
%K nonn
%O 1,6
%A _Pierre CAMI_, Mar 30 2017
%E Missing a(9153)-a(9163) in b-file inserted by _Andrew Howroyd_, Feb 27 2018
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