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%I #12 Aug 03 2022 11:45:37
%S 7,17,29,43,107,163,197,313,457,569,757,827,1303,1487,1783,1997,2339,
%T 2707,2969,3527,3673,3967,4289,4787,5119,5857,7243,9007,9719,10457,
%U 10709,10957,12281,13679,16067,17657,20357,21773,23623,27127,27539,31319,33073,33521,37201,38153,45673,48869,50503
%N Primes p such that if q is the next prime, (p+q)/6 is a triangular number.
%C Primes p such that, if q is the next prime, 1+4*(p+q)/3 is a square.
%H Robert Israel, <a href="/A356293/b356293.txt">Table of n, a(n) for n = 1..10000</a>
%e a(3) = 29 is a term because 29 is prime, the next prime is 31, and (29+31)/6 = 10 is a triangular number.
%p P:= select(isprime,[seq(i,i=3..10^5,2)]):
%p S:= (P[1..-2]+P[2..-1])/3:
%p P[select(i -> S[i]::integer and issqr(1+4*S[i]), [$1..nops(S)])];
%t Select[Partition[Prime[Range[5000]], 2, 1], IntegerQ[Sqrt[4*Total[#]/3 + 1]] &][[;;,1]] (* _Amiram Eldar_, Aug 03 2022 *)
%Y Cf. A000217.
%K nonn
%O 1,1
%A _J. M. Bergot_ and _Robert Israel_, Aug 02 2022
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