%I #38 Jan 05 2020 07:18:35
%S 41,344231,1464251,9646271,48691151,53544461,58182011,68632121,
%T 74656931,74752571,92195381,122898851,164527151,204214541,224671901,
%U 233766041,234327701,269106731,349373891,396416711,412572851,448517501,513644381,530427071,559946021
%N Primes p such that p+6, p+12, p+18, p+20, p+26, p+32, and p+38 are all primes.
%C 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
%C 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
%C 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]
%D Zak Seidov, posting to the Sequence Fans Mailing List, May 21, 2017.
%D Peter Munn, postings to the Sequence Fans Mailing List, May 22 and 24, 2017.
%H Harvey P. Dale, <a href="/A286915/b286915.txt">Table of n, a(n) for n = 1..400</a>
%e 9646271 is included because 9646277, 9646283, 9646289, 9646291, 9646297, 9646303, and 9646309 are all primes.
%t Select[Prime[Range[10^7]],AllTrue[#+{6,12,18,20,26,32,38},PrimeQ]&] (* The program uses the AllTrue function from Mathematica version 10 *)
%t 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 *)
%K nonn
%O 1,1
%A _Harvey P. Dale_, May 24 2017
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