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

Author

N. J. A. Sloane, Dec 19 2006

Keywords

• nonn

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

Programs

Mathematica

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

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

Formula

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