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

17, 97, 16561, 89041, 2579199841, 3497992081, 5645806321, 21103207681, 428888025121, 686770904161, 2726023770721, 4017427557361, 6831989588161, 6933052766641, 10138513506001, 19387278797041, 23452359542401, 35287577206801, 40057354132561, 62093498771041, 64116963608881

Offset

1,1

Comments

Primes of the form x^4 + y^4 such that q = x^2 + y^2 and p = |y^2 - x^2| are both primes.

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.

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

From Robert G. Wilson v, Feb 26 2017: (Start)

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

{5, 3, 17}

{13, 5, 97}

{181, 19, 16561}

{421, 29, 89041}

{71821, 379, 2579199841}

{83641, 409, 3497992081}

{106261, 461, 5645806321}

{205441, 641, 21103207681}

{926161, 1361, 428888025121}

{1171981, 1531, 686770904161}

(End)

Links

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000 (terms 1..59 from Robert G. Wilson v)

Formula

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

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

Example

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

Mathematica

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 *)

Prog

(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

Crossrefs

Subsequence of A002645 and of A094407.

Cf. A103739, A128780.

Sequence in context: A331877 A358572 A262207 * A231667 A044268 A044649

Adjacent sequences: A282994 A282995 A282996 * A282998 A282999 A283000

Keyword

nonn

Author

Thomas Ordowski and Altug Alkan, Feb 26 2017

Extensions

a(11) onward from Robert G. Wilson v, Feb 26 2017

Status

approved