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

%I #13 Nov 16 2021 05:11:00

%S 2,3,7,13,43,127,211,293,743,757,797,811,1429,1597,1721,2087,2113,

%T 2239,2269,2297,2381,2423,2647,3079,3121,3221,3863,4229,4271,4957,

%U 5209,5333,5923,6299,6691,7127,7237,7349,7757,7853,8329,8513,8539,8807,9127,9311,9631,9661

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

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

%C 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 Amiram Eldar, <a href="/A349327/b349327.txt">Table of n, a(n) for n = 1..10000</a>

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

%o (Python)

%o from itertools import islice

%o from sympy import isprime, nextprime

%o def A349327(): # generator of terms

%o n = 2

%o while True:

%o if isprime(n**2-2) and isprime (2*n**2-1): yield n

%o n = nextprime(n)

%o A349327_list = list(islice(A349327(),20)) # _Chai Wah Wu_, Nov 15 2021

%Y Cf. A013929, A023204, A179993.

%Y Intersection of A062326 and A106483.

%Y The prime terms of A225098.

%K nonn

%O 1,1

%A _Amiram Eldar_, Nov 15 2021