%I
%S 2,2,2,5,4,2,5,196,14,129,424,484,22,5164,7726,13325,96873,192098
%N a(n) = least k such that k^(2^n)*(k^(2^n)1)+1 is prime.
%C Numbers of the form k^m*(k^m1)+1 with m > 0, k > 1 may be primes only if m is 3smooth, because these numbers are Phi(6,k^m) and cyclotomic factorizations apply to any prime divisors >3. This series is a subset of A205506 with only m=2^n.
%C Trivially, a(n) <= a(n+1)^2. This upper bound, indeed, holds for a(4) = a(5)^2, a(7) = a(8)^2 and a(11) = a(12)^2.
%C The numbers of this form are Generalized Unique primes (see Links section).
%C a(16)=96873 corresponds to a prime with 653552 decimal digits.
%C The search for a(17) which corresponds to a 1385044decimal digit prime was performed on a small Amazon EC2 cloud farm (40 GRID K520 GPUs), at a cost of approximately $1000 over three weeks.
%C a(18)<=712012 corresponds to a prime with 3068389 decimal digits (not all lower candidates have been checked).  _Serge Batalov_, Jan 15 2018
%C a(19)<=123447 corresponds to a prime with 5338805 decimal digits (not all lower candidates have been checked).  _Serge Batalov_, Jan 15 2018
%H C. Caldwell, <a href="http://primes.utm.edu/top20/page.php?id=44">Generalized unique primes</a>
%t Table[SelectFirst[Range@ 200, PrimeQ[#^(2^n) (#^(2^n)  1) + 1] &], {n, 0, 9}] (* _Michael De Vlieger_, Jan 15 2018 *)
%o (PARI)
%o a(n)=k=1;while(!ispseudoprime(k^(2^n)*(k^(2^n)1)+1),k++);k
%o n=0;while(n<100,print1(a(n),", ");n++) \\ _Derek Orr_, Aug 14 2014
%Y Cf. A205506, A246120, A246121, A153438, A101406, A153436, A056993, A298206.
%K nonn,more,hard
%O 0,1
%A _Serge Batalov_, Aug 14 2014
%E a(16) from _Serge Batalov_, Dec 30 2014
%E a(17) from _Serge Batalov_, Feb 10 2015
