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a(n) is the least A for which there exists B with 0 < B < A so that A^(2^k) + B^(2^k) is prime for k = 0, 1, ..., n.
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%I #20 Aug 28 2022 04:28:36

%S 2,2,2,2,2,2669,34559,26507494,3242781025

%N a(n) is the least A for which there exists B with 0 < B < A so that A^(2^k) + B^(2^k) is prime for k = 0, 1, ..., n.

%C For n < 5, the corresponding primes are Fermat primes, for higher n so-called generalized Fermat primes.

%H Yves Gallot, <a href="https://github.com/galloty/GFP/tree/master/xgfp8">xgfp8</a>, software for calculating this sequence.

%e For n=5, the six numbers 2669 + 720, 2669^2 + 720^2, 2669^4 + 720^4, 2669^8 + 720^8, 2669^16 + 720^16, and 2669^32 + 720^32 are all prime, and (A,B) = (2669,720) is the least pair with this property, so a(5)=2669.

%e For n=6, (A,B) = (34559,29000).

%e For n=7, (A,B) = (26507494,6329559).

%e For n=8, (A,B) = (3242781025,1554825312).

%o (PARI) a(n)=for(A=1, oo, for(B=1, A-1, for(k=0, n, !ispseudoprime(A^(2^k)+B^(2^k)) && next(2)); return(A)))

%Y Cf. A090872, A111635, A291944, A343120.

%K nonn,more,hard

%O 0,1

%A _Jeppe Stig Nielsen_, Apr 05 2021

%E a(7) from _Kellen Shenton_, May 28 2022

%E a(8) from _Kellen Shenton_, Aug 27 2022