OFFSET
1,1
COMMENTS
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)
LINKS
Robert Israel, Table of n, a(n) for n = 1..10000
EXAMPLE
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.
MAPLE
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
MATHEMATICA
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 *)
PROG
(PARI) 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
CROSSREFS
KEYWORD
nonn
AUTHOR
Philip Mizzi, Aug 09 2019
EXTENSIONS
More terms from Michel Marcus, Aug 09 2019
STATUS
approved