A124586 Primes p such that q-p >= 14, where q is the next prime after p.

113, 293, 317, 523, 773, 839, 863, 887, 953, 1069, 1129, 1259, 1327, 1381, 1409, 1583, 1637, 1669, 1759, 1831, 1847, 1913, 1933, 1951, 2039, 2113, 2161, 2179, 2221, 2251, 2311, 2357, 2423, 2477, 2503, 2557, 2593, 2633, 2753, 2803, 2819, 2861, 2939, 2971

Offset

1,1

Comments

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

Links

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

K. Soundararajan, Small gaps between prime numbers: the work of Goldston-Pintz-Yildirim, Bull. Amer. Math. Soc., 44 (2007), 1-18.

Index entries for primes, gaps between

Formula

a(n) = n log n + O(n log^2 n). - Charles R Greathouse IV, May 05 2017

Mathematica

Select[Partition[Prime@ Range@ 430, 2, 1], First@ Differences@ # >= 14 &][[All, 1]] (* Michael De Vlieger, May 12 2017 *)

Prog

(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

Crossrefs

Sequence in context: A001134 A070188 A059331 * A051110 A031932 A216310

Adjacent sequences: A124583 A124584 A124585 * A124587 A124588 A124589

Keyword

nonn

Author

N. J. A. Sloane, Dec 19 2006

Status

approved