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

1,1

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)}.

Links

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

Example

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.

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 *)

Prog

(Sage) [p for p in prime_range(10000) if is_prime((p^2-3)//2) and is_prime((p^2+1)//2)]
(PARI) isok(p) = isprime(p) && isprime((p^2-3)/2) && isprime((p^2+1)/2); \\ Michel Marcus, Mar 31 2017

Crossrefs

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

Sequence in context: A172438 A023233 A048161 * A051642 A007671 A259589

Adjacent sequences: A284031 A284032 A284033 * A284035 A284036 A284037

Keyword

nonn

Author

Giuseppe Coppoletta, Mar 19 2017

Status

approved