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A355188 Primes p such that (2^p+p^2)/3 is prime.

%I A355188 #24 Oct 31 2024 16:26:16
%S A355188 5,7,17,43,61,73,241,739,1297,4211,98519
%N A355188 Primes p such that (2^p+p^2)/3 is prime.
%C A355188 Intersection with A242929 (primes p such that 2^p-p^2 is prime) includes 5, 7 and 17.  Any others?
%C A355188 a(12) > 4*10^5. - _Michael S. Branicky_, Oct 31 2024
%e A355188 a(3) = 17 is a term because (2^17+17^2)/3 = 43787 is prime.
%p A355188 filter:= proc(p) isprime(p) and isprime((2^p+p^2)/3) end proc:
%p A355188 select(filter, [seq(i,i=5..10000,2)]);
%t A355188 Select[Prime[Range[600]], PrimeQ[(2^# + #^2)/3] &] (* _Amiram Eldar_, Jun 23 2022 *)
%o A355188 (PARI) isok(p) = if (isprime(p), my(q=(2^p+p^2)/3); (denominator(q)==1) && ispseudoprime(q)); \\ _Michel Marcus_, Jun 23 2022
%o A355188 (Python)
%o A355188 from itertools import islice
%o A355188 from sympy import isprime, nextprime
%o A355188 def agen():
%o A355188     p = 2
%o A355188     while True:
%o A355188         t = 2**p+p**2
%o A355188         if t%3 == 0 and isprime(t//3):
%o A355188             yield p
%o A355188         p = nextprime(p)
%o A355188 print(list(islice(agen(), 10))) # _Michael S. Branicky_, Jun 23 2022
%Y A355188 Cf. A242929.
%K A355188 nonn,more
%O A355188 1,1
%A A355188 _J. M. Bergot_ and _Robert Israel_, Jun 23 2022
%E A355188 a(11) from _Daniel Suteu_, Jun 25 2022