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

1,1

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.

Links

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

Example

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.

Mathematica

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

Prog

(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

Crossrefs

Cf. A013929, A023204, A179993.

Intersection of A062326 and A106483.

The prime terms of A225098.

Sequence in context: A096263 A007996 A206579 * A166945 A257393 A273814

Adjacent sequences: A349324 A349325 A349326 * A349328 A349329 A349330

Keyword

nonn

Author

Amiram Eldar, Nov 15 2021

Status

approved