A282997 - OEIS

%I #59 Mar 07 2017 00:27:53

%S 17,97,16561,89041,2579199841,3497992081,5645806321,21103207681,

%T 428888025121,686770904161,2726023770721,4017427557361,6831989588161,

%U 6933052766641,10138513506001,19387278797041,23452359542401,35287577206801,40057354132561,62093498771041,64116963608881

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

%C 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 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 Primes of the form x^4 + y^4 such that |y^4 - x^4| is a semiprime.

%C From _Robert G. Wilson v_, Feb 26 2017: (Start)

%C {q, p, a(n) = (p^2+q^2)/2}

%C {5, 3, 17}

%C {13, 5, 97}

%C {181, 19, 16561}

%C {421, 29, 89041}

%C {71821, 379, 2579199841}

%C {83641, 409, 3497992081}

%C {106261, 461, 5645806321}

%C {205441, 641, 21103207681}

%C {926161, 1361, 428888025121}

%C {1171981, 1531, 686770904161}

%C (End)

%H 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 a(n) = A128780(n)^4 + (A128780(n)+1)^4.

%F a(n) == 1 (mod 16).

%e 17 = (3^2 + 5^2)/2 and 5^2 - 3^2 = 4^2.

%t 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 (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 Subsequence of A002645 and of A094407.

%Y Cf. A103739, A128780.

%K nonn

%O 1,1

%A _Thomas Ordowski_ and _Altug Alkan_, Feb 26 2017

%E a(11) onward from _Robert G. Wilson v_, Feb 26 2017