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%I #25 Jan 12 2020 10:47:13
%S 113,293,317,523,773,839,863,887,953,1069,1129,1259,1327,1381,1409,
%T 1583,1637,1669,1759,1831,1847,1913,1933,1951,2039,2113,2161,2179,
%U 2221,2251,2311,2357,2423,2477,2503,2557,2593,2633,2753,2803,2819,2861,2939,2971
%N Primes p such that q-p >= 14, where q is the next prime after p.
%C Also, primes for which residue of (p-1)!+1 modulo p+d equals 1 if d=2,4,6,8,10 and 12. It is evident that all terms p in this sequence have that property, since p+d is composite for d in D = {2, 4, 6, 8, 10, 12}, and so with the least prime q dividing p+d, q <= (p+d)/q <= (p+d)/2 <= (p+12)/2 < p for p > 12 (smaller primes can easily be checked), so q divides (p-1)!. Hence it suffices to show that all p having that property are in this sequence. If not, then p+d is prime but p+d divides (p-1)!, a contradiction. - _Charles R Greathouse IV_, May 05 2017
%H Charles R Greathouse IV, <a href="/A124586/b124586.txt">Table of n, a(n) for n = 1..10000</a>
%H K. Soundararajan, <a href="http://dx.doi.org/10.1090/S0273-0979-06-01142-6">Small gaps between prime numbers: the work of Goldston-Pintz-Yildirim</a>, Bull. Amer. Math. Soc., 44 (2007), 1-18.
%H <a href="/index/Pri#gaps">Index entries for primes, gaps between</a>
%F a(n) = n log n + O(n log^2 n). - _Charles R Greathouse IV_, May 05 2017
%t Select[Partition[Prime@ Range@ 430, 2, 1], First@ Differences@ # >= 14 &][[All, 1]] (* _Michael De Vlieger_, May 12 2017 *)
%o (PARI) is(n)=isprime(n) && !isprime(n+2) && !isprime(n+4) && !isprime(n+6) && !isprime(n+8) && !isprime(n+10) && !isprime(n+12) && n>2 \\ _Charles R Greathouse IV_, Sep 14 2015
%K nonn
%O 1,1
%A _N. J. A. Sloane_, Dec 19 2006
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