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%I #67 Jan 02 2020 13:16:12
%S 251,1741,3301,5101,5381,6311,6361,12641,13451,14741,15791,15901,
%T 17471,18211,19471,23321,26171,30091,30631,53611,56081,62201,63691,
%U 71341,75521,77551,78791,80911,82781,83431,84431,89101,89381,91291,94421
%N Initial prime in set of 4 consecutive primes with common difference 6.
%C Primes p such that p, p+6, p+12, p+18 are consecutive primes.
%C It is conjectured that there exist arbitrarily long sequences of consecutive primes in arithmetic progression. As of March 2013 the record is 10 primes.
%C Note that the Green and Tao reference is about arithmetic progressions that are not necessarily consecutive. - _Michael B. Porter_, Mar 05 2013
%C Subsequence of A023271. - _R. J. Mathar_, Nov 04 2006
%C All terms p == 1 (mod 10) and hence p+24 are always divisible by 5. - _Zak Seidov_, Jun 20 2015
%C Subsequence of A054800, with which is coincides up to a(24), but a(25) = A054800(26). - _M. F. Hasler_, Oct 26 2018
%H T. D. Noe, <a href="/A033451/b033451.txt">Table of n, a(n) for n = 1..1000</a>
%H Jens Kruse Andersen, <a href="http://primerecords.dk/cpap.htm">The Largest Known CPAP's</a>
%H Ben Green and Terence Tao, <a href="http://arxiv.org/abs/math/0404188">The primes contain arbitrarily long arithmetic progressions</a>, arXiv:math/0404188 [math.NT], 2004-2007.
%H B. Green and T. Tao, <a href="http://dx.doi.org/10.4007/annals.2008.167.481">The primes contain arbitrarily long arithmetic progressions</a>, Annals of Math. 167(2008), 481-547.
%H OEIS wiki, <a href="/wiki/Consecutive_primes_in_arithmetic_progression#CPAP_with_given_gap">Consecutive primes in arithmetic progression: CPAP with given gap</a>, updated Jan. 2020
%H <a href="/index/Pri#primes_AP">Index entries for sequences related to primes in arithmetic progressions</a>
%F a(n) = A000040(A090832(n)). - _Zak Seidov_, Jun 20 2015
%e 251, 257, 263, 269 are consecutive primes: 257 = 251 + 6, 263 = 251 + 12, 269 = 251 + 18.
%p N:=10^5: # to get all terms <= N.
%p Primes:=select(isprime,[seq(i,i=3..N+18,2)]):
%p Primes[select(t->[Primes[t+1]-Primes[t], Primes[t+2]-Primes[t+1],
%p Primes[t+3]-Primes[t+2]]=[6,6,6], [$1..nops(Primes)-3])]; # _Muniru A Asiru_, Aug 04 2017
%t A033451 = Reap[ For[p = 2, p < 100000, p = NextPrime[p], p2 = NextPrime[p]; If[p2 - p == 6, p3 = NextPrime[p2]; If[p3 - p2 == 6, p4 = NextPrime[p3]; If[p4 - p3 == 6, Sow[p]]]]]][[2, 1]] (* _Jean-François Alcover_, Jun 28 2012 *)
%t Transpose[Select[Partition[Prime[Range[16000]],4,1],Union[ Differences[ #]] == {6}&]][[1]] (* _Harvey P. Dale_, Jun 17 2014 *)
%o (PARI) p=2;q=3;r=5;forprime(s=7,1e4,if(s-p==18 && s-q==12 && s-r==6, print1(p", ")); p=q;q=r;r=s) \\ _Charles R Greathouse IV_, Feb 14 2013
%Y Intersection of A054800 and A023271.
%Y Cf. A090832, A090833, A090834, A090835, A090836, A090837, A090838, A090839.
%Y Analogous sequences [with common difference in square brackets]: A033447 [12], A033448 [18], A052242 [24], A052243 [30], A058252 [36], A058323 [42], A067388[48].
%Y Cf. A058362, A059044.
%Y Subsequence of A047948.
%K nonn
%O 1,1
%A _Jeff Burch_