%I #67 Aug 28 2022 04:09:25

%S 2,2,2,2,2,9,11,27,14,13,47,22,53,72,216,260,124,1196,200

%N a(n) is the least A for which there exists B with 0 < B < A so that A^(2^n) + B^(2^n) is prime.

%C A^(2^n) + B^(2^n) is called an (extended) generalized Fermat prime, and often denoted F_n(A, B); or xGF(n, A, B).

%C If we require B=1, we get A056993. Therefore a(n) <= A056993(n).

%H Jeppe Stig Nielsen, <a href="/A291944/a291944_1.txt">List of [n, A, B] tuples for this sequence.</a>

%H Chris K. Caldwell, <a href="https://primes.utm.edu/primes/page.php?id=109871">The Prime Database: 72^8192 + 43^8192</a> (related to a(13)).

%H Henri Lifchitz & Renaud Lifchitz, <a href="http://www.primenumbers.net/prptop/searchform.php?form=1196%5E131072%2B595%5E131072">1196^131072+595^131072</a>, a(17).

%H Henri Lifchitz & Renaud Lifchitz, <a href="http://www.primenumbers.net/prptop/searchform.php?form=200%5E262144%2B119%5E262144">200^262144+119^262144</a>, a(18).

%e a(10)=47 corresponds to the prime number 47^1024 + 26^1024, the smallest prime number of the form A^1024 + B^1024 (or more precisely, it minimizes A).

%e a(14)=216 corresponds to the prime number 216^16384 + 109^16384, a 38248-decimal digit PRP, the smallest prime number of the form A^16384 + B^16384. - _Serge Batalov_, Mar 16 2018

%t f[n_] := Monitor[ Block[{a = 2, b}, While[a < Infinity, b = 1 +Mod[a, 2]; While[b < a, If[ PrimeQ[a^2^n + b^2^n], Goto[fini]]; b+=2]; a++]; Label[fini]; {a, b}], {a, b}]; Array[f, 14, 0] (* _Robert G. Wilson v_, Mar 10 2018 *)

%o (PARI) for(n=0,30,for(a=2,10^100,forstep(b=(a % 2)+1,a-1,2,if(ispseudoprime(a^(2^n)+b^(2^n)),print1(a,", ");next(3)))))

%Y Cf. A056993, A253633, A111635.

%K nonn,hard,more

%O 0,1

%A _Jeppe Stig Nielsen_, Mar 09 2018

%E a(14) = 216 (and B = 109) from _Serge Batalov_, Mar 16 2018

%E a(15) = 260 (and B = 179) from _Serge Batalov_, Mar 16 2018

%E a(16) = 124 (and B = 57) from _Serge Batalov_, Mar 16 2018

%E a(17) = 1196 (and B = 595) from _Kellen Shenton_, Aug 10 2022

%E a(18) = 200 (and B = 119) from _Kellen Shenton_, Aug 27 2022