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Primes p such that 2*p^2 - 7, 2*p^2 - 1, and 2*p^2 + 3 are prime.
1

%I #14 Sep 05 2022 12:43:16

%S 43,127,197,3613,3767,4957,28687,29723,40193,46817,66403,78737,89137,

%T 93253,104243,105337,105673,110543,114113,123397,127247,145963,148303,

%U 168713,173293,190387,201893,207367,213613,241597,256117,261323,268253,278543,283807,333227,339373,340913,356173,359143

%N Primes p such that 2*p^2 - 7, 2*p^2 - 1, and 2*p^2 + 3 are prime.

%C All terms == 3 or 7 (mod 10).

%C All terms == 1 or 13 (mod 14). - _Jon E. Schoenfield_, Sep 05 2022

%H Robert Israel, <a href="/A356510/b356510.txt">Table of n, a(n) for n = 1..1000</a>

%e a(3) = 197 is a term because 197, 2*197^2 - 7 = 77611, 2*197^2 - 1 = 77617, and 2*197^2 + 3 = 77621 are all prime.

%p filter:= p -> isprime(p) and isprime(2*p^2+3) and isprime(2*p^2-1) and isprime(2*p^2-7):

%p select(filter, [seq(i,i=3..1000000,2)]);

%t Select[Prime[Range[30000]], AllTrue[2*#^2 + {-7, -1, 3}, PrimeQ] &] (* _Amiram Eldar_, Aug 09 2022 *)

%o (Python)

%o from sympy import isprime

%o def ok(n): return isprime(n) and all(isprime(2*n*n-i) for i in [7, 1, -3])

%o print([k for k in range(10**6) if ok(k)]) # _Michael S. Branicky_, Aug 09 2022

%Y Contained in A106483 and A243595.

%K nonn

%O 1,1

%A _J. M. Bergot_ and _Robert Israel_, Aug 09 2022