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A301738 a(n) is the least A for which there exists B with 0 < B < A so that (A^(2^n) + B^(2^n))/2 is prime.

%I A301738 #29 Apr 01 2021 14:55:22
%S A301738 3,3,3,5,3,3,3,49,7,35,67,75,157,107,71,137,275,531
%N A301738 a(n) is the least A for which there exists B with 0 < B < A so that (A^(2^n) + B^(2^n))/2 is prime.
%C A301738 Both A and B will be odd, with gcd(A, B) = 1. B values can be seen in link section.
%C A301738 If we require B=1, we get A275530. Therefore a(n) <= A275530(n).
%H A301738 H. Lifchitz and R. Lifchitz, <a href="http://www.primenumbers.net/prptop/searchform.php?form=%28275%5E65536%2B53%5E65536%29%2F2">PRP Top (275^65536+53^65536)/2</a>.
%H A301738 H. Lifchitz and R. Lifchitz, <a href="http://www.primenumbers.net/prptop/searchform.php?form=%28531%5E131072%2B25%5E131072%29%2F2">PRP Top (531^131072+25^131072)/2</a>.
%H A301738 Jeppe Stig Nielsen, <a href="/A301738/a301738_1.txt">First primes of the form (A^(2^n) + B^(2^n))/2</a> (gives B values).
%e A301738 a(10)=67 corresponds to the prime number (67^1024 + 57^1024)/2, the smallest prime number of the form (A^1024 + B^1024)/2 (or more precisely, it minimizes A).
%t A301738 Table[a=1; While[Or@@PrimeQ[Table[(a^(2^n)+b^(2^n))/2,{b,a++}]]==False];a,{n,0,9}] (* _Giorgos Kalogeropoulos_, Mar 31 2021 *)
%o A301738 (PARI) for(n=0, 30, forstep(a=3, +oo, 2, forstep(b=1, a-2, 2, if(ispseudoprime((a^(2^n)+b^(2^n))/2), print1(a, ", "); next(3)))))
%o A301738 (Python)
%o A301738 from sympy import isprime
%o A301738 def a(n):
%o A301738   A, p, Ap = 3, 2**n, 3**(2**n)
%o A301738   while True:
%o A301738     if any(isprime((Ap + B**p)//2) for B in range(1, A, 2)): return A
%o A301738     A += 2; Ap = A**p
%o A301738 print([a(n) for n in range(10)]) # _Michael S. Branicky_, Mar 03 2021
%Y A301738 Cf. A291944, A275530.
%K A301738 nonn,hard,more
%O A301738 0,1
%A A301738 _Jeppe Stig Nielsen_, Mar 26 2018
%E A301738 a(16)=275 with B=53, calculated by _Kellen Shenton_, added by _Jeppe Stig Nielsen_, Nov 10 2020
%E A301738 a(17)=531 with B=25, by _Kellen Shenton_, added by _Jeppe Stig Nielsen_, Mar 30 2021