A284035 -id:A284035

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A284034

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

+10
3

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

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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.

KEYWORD

nonn

AUTHOR

Giuseppe Coppoletta, Mar 19 2017

STATUS

approved

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