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

1,1

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

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

Formula

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

Example

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

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 *)

Prog

(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

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.

Sequence in context: A215607 A099734 A054800 * A201793 A183840 A340636

Adjacent sequences: A033448 A033449 A033450 * A033452 A033453 A033454

Keyword

nonn

Author

Jeff Burch

Status

approved