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%I M3760 #40 Apr 07 2022 10:44:53
%S 5,7,11,13,17,37,41,67,97,101,103,107,191,193,223,227,277,307,311,347,
%T 457,461,613,641,821,823,853,857,877,881,1087,1091,1277,1297,1301,
%U 1423,1427,1447,1481,1483,1487,1607,1663,1693,1783,1867,1871,1873,1993,1997
%N Prime triples: p; p+2 or p+4; p+6 all prime.
%C Or, prime(m) such that prime(m+2) = prime(m)+6. - _Zak Seidov_, May 07 2012
%D H. Riesel, "Prime numbers and computer methods for factorization", Progress in Mathematics, Vol. 57, Birkhauser, Boston, 1985, Chap. 4, see p. 65.
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H Zak Seidov, <a href="/A007529/b007529.txt">Table of n, a(n) for n = 1..10000</a>
%H Ernest G. Hibbs, <a href="https://www.proquest.com/openview/4012f0286b785cd732c78eb0fc6fce80">Component Interactions of the Prime Numbers</a>, Ph. D. Thesis, Capitol Technology Univ. (2022), see p. 33.
%F a(n) = A098415(n) - 6. - _Zak Seidov_, Apr 30 2015
%p N:= 10000: # to get all terms <= N
%p Primes:= select(isprime, [seq(2*i+1, i=1..floor((N+5)/2))]):locs:= select(t -> Primes[t+2]-Primes[t]=6, [$1..nops(Primes)-2]):
%p Primes[locs]; # _Robert Israel_, Apr 30 2015
%t ptrsQ[n_]:=PrimeQ[n+6]&&(PrimeQ[n+2]||PrimeQ[n+4])
%t Select[Prime[Range[400]],ptrsQ] (* _Harvey P. Dale_, Mar 08 2011 *)
%o (Magma) [NthPrime(n): n in [1..310] | (NthPrime(n)+6) eq NthPrime(n+2)]; // _Bruno Berselli_, May 07 2012
%o (PARI) p=2;q=3;forprime(r=5,1e4,if(r-p==6,print1(p", "));p=q;q=r) \\ _Charles R Greathouse IV_, May 07 2012
%Y Cf. A031924, A023201, A098414, A098415, A098424, A098416, A098417.
%Y Union of A022004 and A022005.
%K nonn
%O 1,1
%A _N. J. A. Sloane_, _Robert G. Wilson v_
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