A002645 -id:A002645 - cp's OEIS frontend

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

Thomas Ordowski and Altug Alkan, Feb 26 2017

nonn

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)

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

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

Subsequence of A002645 and of A094407.

Cf. A103739, A128780.

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 *)

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

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

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

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

A140834 Primes that are the sum of at most four nonzero 4th powers.

Original entry on oeis.org

2, 3, 17, 19, 83, 97, 113, 163, 179, 257, 337, 353, 419, 499, 593, 641, 643, 673, 769, 787, 881, 883, 1153, 1297, 1409, 1459, 1553, 1889, 2003, 2083, 2131, 2417, 2579, 2593, 2609, 2657, 2659, 2689, 2819, 3169, 3217, 3697, 3779, 3889, 3907, 4099, 4129, 4177

Offset: 1

Jonathan Vos Post, Jul 18 2008

This sequence was checked by T. D. Noe, who had supplied the b-list for A004833. A037896 is a subset of {Primes that are the sum of at exactly 2 nonzero 4th powers}, itself a subset of A002645 Quartan primes: primes of the form x^4 + y^4, x>0, y>0.

Cf. A000040, A000583, A002645, A003294, A004833, A037896, A085318, A122540, A126657, A133740.

A000040 INTERSECTION A004833. {A133740 = Primes that are the sum of at exactly 4 nonzero 4th powers} UNION {A085318 = Primes that are the sum of at exactly 3 nonzero 4th powers} UNION {A002645 = Primes that are the sum of at exactly 2 nonzero 4th powers}.

Missing term 353 inserted by Georg Fischer, May 11 2024

cp's OEIS Frontend

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

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