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

%I A349327 #13 Nov 16 2021 05:11:00
%S A349327 2,3,7,13,43,127,211,293,743,757,797,811,1429,1597,1721,2087,2113,
%T A349327 2239,2269,2297,2381,2423,2647,3079,3121,3221,3863,4229,4271,4957,
%U A349327 5209,5333,5923,6299,6691,7127,7237,7349,7757,7853,8329,8513,8539,8807,9127,9311,9631,9661
%N A349327 Primes p such that 2*p^2 is a term of A179993.
%C A349327 The numbers of the form 2*p^2 where p is a term of this sequence are the only nonsquarefree terms of A179993.
%C A349327 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.
%H A349327 Amiram Eldar, <a href="/A349327/b349327.txt">Table of n, a(n) for n = 1..10000</a>
%e A349327 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.
%e A349327 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.
%t A349327 q[n_] := AllTrue[{n, n^2 - 2, 2*n^2 - 1}, PrimeQ]; Select[Range[10000], q]
%o A349327 (Python)
%o A349327 from itertools import islice
%o A349327 from sympy import isprime, nextprime
%o A349327 def A349327(): # generator of terms
%o A349327     n = 2
%o A349327     while True:
%o A349327         if isprime(n**2-2) and isprime (2*n**2-1): yield n
%o A349327         n = nextprime(n)
%o A349327 A349327_list = list(islice(A349327(),20)) # _Chai Wah Wu_, Nov 15 2021
%Y A349327 Cf. A013929, A023204, A179993.
%Y A349327 Intersection of A062326 and A106483.
%Y A349327 The prime terms of A225098.
%K A349327 nonn
%O A349327 1,1
%A A349327 _Amiram Eldar_, Nov 15 2021