A349327 Primes p such that 2*p^2 is a term of A179993.

2, 3, 7, 13, 43, 127, 211, 293, 743, 757, 797, 811, 1429, 1597, 1721, 2087, 2113, 2239, 2269, 2297, 2381, 2423, 2647, 3079, 3121, 3221, 3863, 4229, 4271, 4957, 5209, 5333, 5923, 6299, 6691, 7127, 7237, 7349, 7757, 7853, 8329, 8513, 8539, 8807, 9127, 9311, 9631, 9661

Offset: 1

Author

Amiram Eldar, Nov 15 2021

Keywords

• nonn

Comments

The numbers of the form 2*p^2 where p is a term of this sequence are the only nonsquarefree terms of A179993.

Equivalently, primes p such that p^2 - 2 and 2*p^2 - 1 are also primes, or primes p such that p^2 - 2 is a term of A023204.

Examples

			2 is a term since 2*2^2 = 8 = 1*8 = 2*4 is a term of A179993: 8 - 1 = 7 and 4 - 2 = 2 are both primes.
3 is a term since 2*3^2 = 18 = 1*18 = 2*9 = 3*6 is a term of A179993: 18 - 1 = 17, 9 - 2 = 7 and 6 - 3 = 3 are all primes.
		

Links

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

Crossrefs

Cf. A013929, A023204, A179993.

Intersection of A062326 and A106483.

The prime terms of A225098.

Programs

Mathematica

q[n_] := AllTrue[{n, n^2 - 2, 2*n^2 - 1}, PrimeQ]; Select[Range[10000], q]

Python

from itertools import islice
from sympy import isprime, nextprime
def A349327(): # generator of terms
    n = 2
    while True:
        if isprime(n**2-2) and isprime (2*n**2-1): yield n
        n = nextprime(n)
A349327_list = list(islice(A349327(),20)) # Chai Wah Wu, Nov 15 2021