A130621 -id:A130621 - cp's OEIS frontend

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

13, 37, 277, 613, 12157, 14557, 23053, 55213, 81013, 203317, 331333, 393853, 824773, 867253, 1008037, 2038573, 3026053, 3322213, 5198197, 5497237, 5793517, 5984053, 9107173, 17246413, 20850757, 20871853, 21327997, 25363573, 25678573, 27258613, 29134597, 30153037, 33313333

Offset: 1

Philip Mizzi, Aug 09 2019

nonn

A prime triple is a set of three prime numbers of the form (p, p+2, p+6) or (p, p+4, p+6).
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.
PROOF:
From Bernard Schott, Aug 09 2019: (Start)
If a == 0 (mod 3) ==> a^2 == 0 (mod 3),
If a == 1 (mod 3) ==> a^2 == 1 (mod 3),
If a == 2 (mod 3) ==> a^2 == 4 == 1 (mod 3).
Hence, a square is always == 0 or == 1 (mod 3)
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)

			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.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A130621.

Intersection of A022005 and A206279.

Res:= NULL: count:= 0:
for k from 0 while count < 100 do
  for x in [6*k+1,6*k+5] do
    p:= (x^2-10)/3;
    if isprime(p) and isprime(p+4) and isprime(p+6) then
      count:= count+1;
      Res:= Res, p
    fi
od od:
Res; # Robert Israel, Aug 13 2019

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 *)

issq(p) = issquare(3*p+10);
istriple(p) = isprime(p+4) && isprime(p+6);
isok(p) = isprime(p) && istriple(p) && issq(p); \\ Michel Marcus, Aug 10 2019

More terms from Michel Marcus, Aug 09 2019

cp's OEIS Frontend

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

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