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%I #12 Apr 03 2023 10:36:13
%S 7,11,13,13,17,19,37,41,43,67,71,73,97,101,103,103,107,109,193,197,
%T 199,223,227,229,277,281,283,307,311,313,457,461,463,613,617,619,823,
%U 827,829,853,857,859,877,881,883,1087,1091,1093,1297,1301,1303,1423,1427,1429
%N Table read by rows: list of prime triples of the form (p, p+4, p+6).
%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 Initial members p of prime triples of the form (p, p+4, p+6) are congruent to 7 or 13 (mod 30).
%C Also called prime triples of the second kind.
%H C. K. Caldwell, Top Twenty page, <a href="https://t5k.org/top20/page.php?id=61">Triplet</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/PrimeTriplet.html">Prime Triplet</a>
%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Prime_triple">Prime triple</a>
%H <a href="/index/Pri#gaps">Index entries for primes, gaps between</a>
%F a(3*n-2) = A022005(n).
%e The table starts:
%e 7, 11, 13;
%e 13, 17, 19;
%e 37, 41, 43;
%e ...
%t Prime@ Range[#, # + 2] &@ PrimePi@ Select[Prime@ Range@ 240, Times @@ Boole@ PrimeQ[# + {4, 6}] > 0 &] // Flatten (* _Michael De Vlieger_, Aug 02 2016 *)
%o (Magma) &cat[[p, p+4, p+6]: p in PrimesUpTo(1423) | p mod 30 in {7, 13} and IsPrime(p+4) and IsPrime(p+6)];
%Y Cf. A022005, A270999, A271000, A275515.
%K nonn,tabf
%O 1,1
%A _Arkadiusz Wesolowski_, Jul 31 2016
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