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A282997 Primes of the form (p^2 + q^2)/2 such that |q^2 - p^2| is square, where p and q are prime.

%I A282997 #59 Mar 07 2017 00:27:53
%S A282997 17,97,16561,89041,2579199841,3497992081,5645806321,21103207681,
%T A282997 428888025121,686770904161,2726023770721,4017427557361,6831989588161,
%U A282997 6933052766641,10138513506001,19387278797041,23452359542401,35287577206801,40057354132561,62093498771041,64116963608881
%N A282997 Primes of the form (p^2 + q^2)/2 such that |q^2 - p^2| is square, where p and q are prime.
%C A282997 Primes of the form x^4 + y^4 such that q = x^2 + y^2 and p = |y^2 - x^2| are both primes.
%C A282997 Primes of the form n^4 + (n+1)^4 such that q = n^2 + (n+1)^2 and p = 2n+1 are both primes; so for n in A128780.
%C A282997 Primes of the form x^4 + y^4 such that |y^4 - x^4| is a semiprime.
%C A282997 From _Robert G. Wilson v_, Feb 26 2017: (Start)
%C A282997 {q, p, a(n) = (p^2+q^2)/2}
%C A282997 {5, 3, 17}
%C A282997 {13, 5, 97}
%C A282997 {181, 19, 16561}
%C A282997 {421, 29, 89041}
%C A282997 {71821, 379, 2579199841}
%C A282997 {83641, 409, 3497992081}
%C A282997 {106261, 461, 5645806321}
%C A282997 {205441, 641, 21103207681}
%C A282997 {926161, 1361, 428888025121}
%C A282997 {1171981, 1531, 686770904161}
%C A282997 (End)
%H A282997 Charles R Greathouse IV, <a href="/A282997/b282997.txt">Table of n, a(n) for n = 1..10000</a> (terms 1..59 from Robert G. Wilson v)
%F A282997 a(n) = A128780(n)^4 + (A128780(n)+1)^4.
%F A282997 a(n) == 1 (mod 16).
%e A282997 17 = (3^2 + 5^2)/2 and 5^2 - 3^2 = 4^2.
%t A282997 lst = {}; a = 2; While[a < 2501, b = Mod[a, 2] + 1; While[b < a, If[ PrimeQ[a^4 + b^4] && PrimeOmega[a^4 - b^4] == 2, AppendTo[lst, (a^4 + b^4)]]; b += 2]; a++]; lst (* _Robert G. Wilson v_, Feb 27 2017 *)
%o A282997 (PARI) list(lim)=my(v=List(),t,n); while((t=n++^4+(n+1)^4)<=lim, if(isprime(t) && isprime(n^2+(n+1)^2) && isprime(2*n+1), listput(v,t))); Vec(v) \\ _Charles R Greathouse IV_, Feb 26 2017
%Y A282997 Subsequence of A002645 and of A094407.
%Y A282997 Cf. A103739, A128780.
%K A282997 nonn
%O A282997 1,1
%A A282997 _Thomas Ordowski_ and _Altug Alkan_, Feb 26 2017
%E A282997 a(11) onward from _Robert G. Wilson v_, Feb 26 2017