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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. 1
 17, 97, 16561, 89041, 2579199841, 3497992081, 5645806321, 21103207681, 428888025121, 686770904161, 2726023770721, 4017427557361, 6831989588161, 6933052766641, 10138513506001, 19387278797041, 23452359542401, 35287577206801, 40057354132561, 62093498771041, 64116963608881 (list; graph; refs; listen; history; text; internal format)
 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: A152913 A184327 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

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Last modified October 17 22:37 EDT 2019. Contains 328134 sequences. (Running on oeis4.)