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A309594 Smallest members of prime triples, the sum of which results in a perfect square.

%I A309594 #45 Aug 13 2019 13:17:27
%S A309594 13,37,277,613,12157,14557,23053,55213,81013,203317,331333,393853,
%T A309594 824773,867253,1008037,2038573,3026053,3322213,5198197,5497237,
%U A309594 5793517,5984053,9107173,17246413,20850757,20871853,21327997,25363573,25678573,27258613,29134597,30153037,33313333
%N A309594 Smallest members of prime triples, the sum of which results in a perfect square.
%C A309594 A prime triple is a set of three prime numbers of the form (p, p+2, p+6) or (p, p+4, p+6).
%C A309594 The smallest prime of the first form of these triples is not part of this sequence because p + (p+2) + (p+6) = 3p +8 and a number of this form is never a square.
%C A309594 PROOF:
%C A309594 From _Bernard Schott_, Aug 09 2019: (Start)
%C A309594 If a == 0 (mod 3) ==> a^2 == 0 (mod 3),
%C A309594 If a == 1 (mod 3) ==> a^2 == 1 (mod 3),
%C A309594 If a == 2 (mod 3) ==> a^2 == 4 == 1 (mod 3).
%C A309594 Hence, a square is always == 0 or == 1 (mod 3)
%C A309594 As p + (p+2) + (p+6) = 3*p+8, and 3*p+8 == 2 (mod 3), there is no prime triple of the form (p, p+2, p+6) whose sum 3*p + 8 can be a square. (End)
%H A309594 Robert Israel, <a href="/A309594/b309594.txt">Table of n, a(n) for n = 1..10000</a>
%e A309594 Let p = 277 (prime), q = p+4 = 281 (prime), r = p+6 = 283 (prime). We now have a prime triple. p+q+r = 841 = 29^2, a perfect square.
%p A309594 Res:= NULL: count:= 0:
%p A309594 for k from 0 while count < 100 do
%p A309594   for x in [6*k+1,6*k+5] do
%p A309594     p:= (x^2-10)/3;
%p A309594     if isprime(p) and isprime(p+4) and isprime(p+6) then
%p A309594       count:= count+1;
%p A309594       Res:= Res, p
%p A309594     fi
%p A309594 od od:
%p A309594 Res; # _Robert Israel_, Aug 13 2019
%t A309594 ok[p_] := If[AllTrue[{p, p+4, p+6}, PrimeQ], Sow@p]; Reap[Do[ok[3 y^2 + 2 y - 3]; ok[3 y^2 + 4 y - 2], {y, 4000}]][[2, 1]] (* _Giovanni Resta_, Aug 09 2019 *)
%o A309594 (PARI) issq(p) = issquare(3*p+10);
%o A309594 istriple(p) = isprime(p+4) && isprime(p+6);
%o A309594 isok(p) = isprime(p) && istriple(p) && issq(p); \\ _Michel Marcus_, Aug 10 2019
%Y A309594 Cf. A130621.
%Y A309594 Intersection of A022005 and A206279.
%K A309594 nonn
%O A309594 1,1
%A A309594 _Philip Mizzi_, Aug 09 2019
%E A309594 More terms from _Michel Marcus_, Aug 09 2019