A309594 - OEIS

%I #45 Aug 13 2019 13:17:27

%S 13,37,277,613,12157,14557,23053,55213,81013,203317,331333,393853,

%T 824773,867253,1008037,2038573,3026053,3322213,5198197,5497237,

%U 5793517,5984053,9107173,17246413,20850757,20871853,21327997,25363573,25678573,27258613,29134597,30153037,33313333

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

%C 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 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 PROOF:

%C From _Bernard Schott_, Aug 09 2019: (Start)

%C If a == 0 (mod 3) ==> a^2 == 0 (mod 3),

%C If a == 1 (mod 3) ==> a^2 == 1 (mod 3),

%C If a == 2 (mod 3) ==> a^2 == 4 == 1 (mod 3).

%C Hence, a square is always == 0 or == 1 (mod 3)

%C 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 Robert Israel, <a href="/A309594/b309594.txt">Table of n, a(n) for n = 1..10000</a>

%e 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 Res:= NULL: count:= 0:

%p for k from 0 while count < 100 do

%p   for x in [6*k+1,6*k+5] do

%p     p:= (x^2-10)/3;

%p     if isprime(p) and isprime(p+4) and isprime(p+6) then

%p       count:= count+1;

%p       Res:= Res, p

%p     fi

%p od od:

%p Res; # _Robert Israel_, Aug 13 2019

%t 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 (PARI) issq(p) = issquare(3*p+10);

%o istriple(p) = isprime(p+4) && isprime(p+6);

%o isok(p) = isprime(p) && istriple(p) && issq(p); \\ _Michel Marcus_, Aug 10 2019

%Y Cf. A130621.

%Y Intersection of A022005 and A206279.

%K nonn

%O 1,1

%A _Philip Mizzi_, Aug 09 2019

%E More terms from _Michel Marcus_, Aug 09 2019