A033451 Initial prime in set of 4 consecutive primes with common difference 6.

251, 1741, 3301, 5101, 5381, 6311, 6361, 12641, 13451, 14741, 15791, 15901, 17471, 18211, 19471, 23321, 26171, 30091, 30631, 53611, 56081, 62201, 63691, 71341, 75521, 77551, 78791, 80911, 82781, 83431, 84431, 89101, 89381, 91291, 94421

Offset: 1

Author

Jeff Burch

Keywords

• nonn

Comments

Primes p such that p, p+6, p+12, p+18 are consecutive primes.

It is conjectured that there exist arbitrarily long sequences of consecutive primes in arithmetic progression. As of March 2013 the record is 10 primes.

Note that the Green and Tao reference is about arithmetic progressions that are not necessarily consecutive. - Michael B. Porter, Mar 05 2013

Subsequence of A023271. - R. J. Mathar, Nov 04 2006

All terms p == 1 (mod 10) and hence p+24 are always divisible by 5. - Zak Seidov, Jun 20 2015

Subsequence of A054800, with which is coincides up to a(24), but a(25) = A054800(26). - M. F. Hasler, Oct 26 2018

Examples

			251, 257, 263, 269 are consecutive primes: 257 = 251 + 6, 263 = 251 + 12, 269 = 251 + 18.
		

Links

• T. D. Noe, Table of n, a(n) for n = 1..1000
• Jens Kruse Andersen, The Largest Known CPAP's
• Ben Green and Terence Tao, The primes contain arbitrarily long arithmetic progressions, arXiv:math/0404188 [math.NT], 2004-2007.
• B. Green and T. Tao, The primes contain arbitrarily long arithmetic progressions, Annals of Math. 167(2008), 481-547.
• OEIS wiki, Consecutive primes in arithmetic progression: CPAP with given gap, updated Jan. 2020
• Index entries for sequences related to primes in arithmetic progressions

Crossrefs

Intersection of A054800 and A023271.

Cf. A090832, A090833, A090834, A090835, A090836, A090837, A090838, A090839.

Analogous sequences [with common difference in square brackets]: A033447 [12], A033448 [18], A052242 [24], A052243 [30], A058252 [36], A058323 [42], A067388[48].

Cf. A058362, A059044.

Subsequence of A047948.

Programs

Maple

N:=10^5: # to get all terms <= N.
Primes:=select(isprime,[seq(i,i=3..N+18,2)]):
Primes[select(t->[Primes[t+1]-Primes[t], Primes[t+2]-Primes[t+1],
Primes[t+3]-Primes[t+2]]=[6,6,6], [$1..nops(Primes)-3])]; # Muniru A Asiru, Aug 04 2017

Mathematica

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 *)
Transpose[Select[Partition[Prime[Range[16000]],4,1],Union[ Differences[ #]] == {6}&]][[1]] (* Harvey P. Dale, Jun 17 2014 *)

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

Formula

a(n) = A000040(A090832(n)). - Zak Seidov, Jun 20 2015