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A124586 Primes p such that q-p >= 14, where q is the next prime after p. 3
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 (list; graph; refs; listen; history; text; internal format)
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

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Last modified April 4 20:47 EDT 2020. Contains 333229 sequences. (Running on oeis4.)