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A286915
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Primes p such that p+6, p+12, p+18, p+20, p+26, p+32, and p+38 are all primes.
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1
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41, 344231, 1464251, 9646271, 48691151, 53544461, 58182011, 68632121, 74656931, 74752571, 92195381, 122898851, 164527151, 204214541, 224671901, 233766041, 234327701, 269106731, 349373891, 396416711, 412572851, 448517501, 513644381, 530427071, 559946021
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OFFSET
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1,1
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COMMENTS
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Many, but not all, of the terms are the smallest prime in a sequence of 8 consecutive primes with first differences equal to 6, 6, 6, 2, 6, 6, 6. -Harvey P. Dale, May 24 2017
Of the first 400 terms, 311 are the smallest prime in a sequence of 8 consecutive primes with first differences equal to 6, 6, 6, 2, 6, 6, 6. - Harvey P. Dale_, May 25 2017
All terms are congruent to 11 (mod 30). - Zak Seidov, May 24 2017 [Proof: Suppose p == x (mod 30). Then all of x, x+6, x+12, x+18, x+20, x+26, x+2, and x+8 must be relatively prime to 30. This implies x = 11. (We cannot have x=1, for then p+2 would be == 3 (mod 30), we cannot have x=7, for then p+18 would be == 25 (mod 30), and so on.) - N. J. A. Sloane, May 24 2017]
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REFERENCES
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Zak Seidov, posting to the Sequence Fans Mailing List, May 21, 2017.
Peter Munn, postings to the Sequence Fans Mailing List, May 22 and 24, 2017.
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LINKS
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EXAMPLE
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9646271 is included because 9646277, 9646283, 9646289, 9646291, 9646297, 9646303, and 9646309 are all primes.
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MATHEMATICA
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Select[Prime[Range[10^7]], AllTrue[#+{6, 12, 18, 20, 26, 32, 38}, PrimeQ]&] (* The program uses the AllTrue function from Mathematica version 10 *)
Select[{41, 251, 311}+#&/@(330*Range[0, 1000000])//Flatten, AllTrue[#+{0, 6, 12, 18, 20, 26, 32, 38}, PrimeQ]&] (* Much faster than the first program above *) (* The program uses the AllTrue function from Mathematica version 10 *) (* Harvey P. Dale, May 24 2017 *)
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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