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A240971 Primes p such that (p^2 + p + 1)/3 is prime.

%I A240971 #21 Feb 16 2025 08:33:21
%S A240971 7,13,19,31,43,73,97,103,127,157,199,223,241,271,409,421,661,673,727,
%T A240971 859,883,937,1021,1039,1051,1063,1093,1447,1483,1609,1657,1669,1723,
%U A240971 1753,1861,1879,1993,2029,2203,2437,2539,2677,2719,2803,2833,2953,3079,3121
%N A240971 Primes p such that (p^2 + p + 1)/3 is prime.
%C A240971 Under Schinzel's hypothesis, there are infinitely many primes of this form.
%C A240971 p must be of form 6k+1 to give an integer. A053182 lists when p^2 + p + 1 is prime. - _Jens Kruse Andersen_, Aug 06 2014
%H A240971 Vincenzo Librandi, <a href="/A240971/b240971.txt">Table of n, a(n) for n = 1..4900</a>
%H A240971 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/SchinzelsHypothesis.html">Schinzel's Hypothesis</a>.
%p A240971 select(n -> isprime(n) and isprime((n^2 + n + 1)/3), [seq(6*k+1,k=1..1000)]); # _Robert Israel_, Aug 05 2014
%t A240971 Select[Prime[Range[500]], PrimeQ[(#^2 + # + 1)/3] &]
%o A240971 (Magma) [p: p in PrimesInInterval(3,3500)| IsPrime((p^2+p+1) div 3)];
%o A240971 (PARI) forprime(p=1,10^4,s=(p^2+p+1)/3;if(floor(s)==s,if(isprime(s),print1(p,", ")))) \\ _Derek Orr_, Aug 05 2014
%Y A240971 Cf. A053182.
%K A240971 nonn
%O A240971 1,1
%A A240971 _Vincenzo Librandi_, Aug 05 2014