A284034 Primes p such that (p^2 - 3)/2 and (p^2 + 1)/2 are twin primes.

3, 5, 11, 19, 29, 79, 101, 349, 409, 449, 521, 569, 571, 661, 739, 991, 1091, 1129, 1181, 1459, 1489, 1531, 1901, 2269, 2281, 2341, 2351, 2389, 2549, 2659, 2671, 2719, 2729, 2731, 3109, 4049, 4349, 5279, 5431, 5471, 5531, 5591, 5669, 6329, 6359, 6871, 7559, 7741

Offset: 1

Author

Giuseppe Coppoletta, Mar 19 2017

Keywords

• nonn

Comments

Primes which correspond to the short leg of an integral right triangle whose hypotenuse is part of a twin prime pair.

Each term p of the sequence must be part of a Pythagorean triple of the form {p, (p^2 - 1)/2, (p^2 + 1)/2} corresponding to {a(n), A284035(n) - 1, A284035(n)}.

Examples

			The prime p = 79 is in the sequence because (p^2-3)/2 = 3119 and (p^2+1)/2 = 3121 are twin primes. Remark that {79, 3120, 3121} is a Pythagorean triple.
		

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A284035, A048161, A165635, A284036, A051642.

Programs

Mathematica

Select[Prime@ Range[10^3], Function[p, Times @@ Boole@ Map[PrimeQ[(p^2 + #)/2 ] &, {-3, 1}] == 1]] (* Michael De Vlieger, Mar 20 2017 *)
Select[Prime[Range[1000]],AllTrue[{(#^2-3)/2,(#^2+1)/2},PrimeQ]&] (* The program uses the AllTrue function from Mathematica version 10 *) (* Harvey P. Dale, Sep 04 2017 *)

PARI

isok(p) = isprime(p) && isprime((p^2-3)/2) && isprime((p^2+1)/2); \\ Michel Marcus, Mar 31 2017

Sage

[p for p in prime_range(10000) if is_prime((p^2-3)//2) and is_prime((p^2+1)//2)]