%I A022004 #133 Jun 11 2026 11:52:57
%S A022004 5,11,17,41,101,107,191,227,311,347,461,641,821,857,881,1091,1277,
%T A022004 1301,1427,1481,1487,1607,1871,1997,2081,2237,2267,2657,2687,3251,
%U A022004 3461,3527,3671,3917,4001,4127,4517,4637,4787,4931,4967,5231,5477
%N A022004 Initial members of prime triples (p, p+2, p+6).
%C A022004 Subsequence of A001359. - _R. J. Mathar_, Feb 10 2013
%C A022004 All terms are congruent to 5 (mod 6). - _Matt C. Anderson_, May 22 2015
%C A022004 Intersection of A001359 and A023201. - _Zak Seidov_, Mar 12 2016
%H A022004 Matt C. Anderson  <a href="/A022004/b022004.txt">Table of n, a(n) for n = 1..10000</a> (terms 1..1000 from T. D. Noe)
%H A022004 T. Forbes and Norman Luhn, <a href="http://www.pzktupel.de/ktuplets">Prime k-tuplets</a>
%H A022004 Norman Luhn, <a href="https://pzktupel.de/smarchive.php">The big database of "The smallest prime k-tuplets" (compressed files)</a>.
%H A022004 R. J. Mathar, <a href="/A022004/a022004_2.pdf">Table of Prime Gap Constellations</a> (2013,2024), 275 pages (no not print...)
%H A022004 Thomas R. Nicely, <a href="https://web.archive.org/web/20191115021905/https://faculty.lynchburg.edu/~nicely/triplets/t3a_0000.htm">Enumeration of the prime triples (q,q+2,q+6) to 1e16</a>.
%H A022004 P. Pollack, <a href="http://www.math.dartmouth.edu/~ppollack/notes.pdf">Analytic and Combinatorial Number Theory</a>, Course Notes, p. 132, ex. 3.4.3. [Broken link]
%H A022004 P. Pollack, <a href="https://web.archive.org/web/20130509125813/http://alpha01.dm.unito.it/personalpages/cerruti/ac/notes.pdf">Analytic and Combinatorial Number Theory</a>, Course Notes, p. 132, ex. 3.4.3.
%H A022004 Maxie D. Schmidt, <a href="https://arxiv.org/abs/1701.04741">New Congruences and Finite Difference Equations for Generalized Factorial Functions</a>, arXiv:1701.04741 [math.CO], 2017.
%H A022004 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/PrimeTriplet.html">Prime Triplet</a>
%H A022004 <a href="/index/Pri#gaps">Index entries for primes, gaps between</a>
%p A022004 A022004 := proc(n)
%p A022004     if n= 1 then
%p A022004         5;
%p A022004     else
%p A022004         for a from procname(n-1)+2 by 2 do
%p A022004             if isprime(a) and isprime(a+2) and isprime(a+6) then
%p A022004                 return a;
%p A022004             end if;
%p A022004         end do:
%p A022004     end if;
%p A022004 end proc: # _R. J. Mathar_, Jul 11 2012
%t A022004 Select[Prime[Range[1000]], PrimeQ[#+2] && PrimeQ[#+6]&] (* _Vladimir Joseph Stephan Orlovsky_, Mar 30 2011 *)
%t A022004 Transpose[Select[Partition[Prime[Range[1000]],3,1],Differences[#]=={2,4}&]][[1]] (* _Harvey P. Dale_, Dec 24 2011 *)
%o A022004 (Magma) [ p: p in PrimesUpTo(10000) | IsPrime(p+2) and IsPrime(p+6) ]; // _Vincenzo Librandi_, Nov 19 2010
%o A022004 (PARI) is(n)=isprime(n)&&isprime(n+2)&&isprime(n+6) \\ _Charles R Greathouse IV_, Jul 01 2013
%o A022004 (Python)
%o A022004 from sympy import primerange
%o A022004 def aupto(limit):
%o A022004   p, q, alst = 2, 3, []
%o A022004   for r in primerange(5, limit+7):
%o A022004     if p+2 == q and p+6 == r: alst.append(p)
%o A022004     p, q = q, r
%o A022004   return alst
%o A022004 print(aupto(5477)) # _Michael S. Branicky_, May 11 2021
%Y A022004 Cf. A073648, A098412, A372247 (subsequence).
%Y A022004 Cf. A001359, A022005, A023201.
%Y A022004 Subsequence of A007529.
%K A022004 nonn,easy
%O A022004 1,1
%A A022004 _Warut Roonguthai_