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A033451
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Initial prime in set of 4 consecutive primes with common difference 6.
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42
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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
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OFFSET
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1,1
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COMMENTS
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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
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REFERENCES
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Ben Green and Terence Tao, "The primes contain arbitrarily long arithmetic progressions", Annals of Mathematics 167:2 (2008), 481-547.
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LINKS
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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
Index entries for sequences related to primes in arithmetic progressions
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EXAMPLE
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251,257,263,269 are consecutive primes: 257=251+6,263=251+12,269=251+18
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MATHEMATICA
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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 *)
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PROG
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(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
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CROSSREFS
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Intersection of A054800 and A023271.
Cf. A090832, A090833, A090834, A090835, A090836, A090837, A090838, A090839.
Analogous sequences [with common difference in square brackets]: A033451 [6], A033447 [12], A033448 [18], A052242 [24], A052243 [30].
Cf. A058362, A059044.
Cf. A047948 (three consecutive primes with difference 6)
Sequence in context: A215607 A099734 A054800 * A201793 A183840 A218639
Adjacent sequences: A033448 A033449 A033450 * A033452 A033453 A033454
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KEYWORD
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nonn,changed
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AUTHOR
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Jeff Burch
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STATUS
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approved
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