A124590 Primes p such that q-p <= 6, where q is the next prime after p.

2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 97, 101, 103, 107, 109, 127, 131, 137, 149, 151, 157, 163, 167, 173, 179, 191, 193, 197, 223, 227, 229, 233, 239, 251, 257, 263, 269, 271, 277, 281, 307, 311, 313, 331, 347, 349, 353, 367

Offset: 1

Author

N. J. A. Sloane, Dec 19 2006

Keywords

• nonn
• easy

Comments

Goldston, Graham, Pintz, & Yilidirm give a conditional proof that this sequence is infinite; see their Theorem 4. - Charles R Greathouse IV, Jul 31 2013

Links

• Charles R Greathouse IV, Table of n, a(n) for n = 1..10000
• D. A. Goldston, S. W. Graham, J. Pintz, C. Y. Yilidirm, Small gaps between primes or almost primes (2005)
• K. Soundararajan, Small gaps between prime numbers: the work of Goldston-Pintz-Yildirim, Bull. Amer. Math. Soc. 44 (2007), pp. 1-18.
• Index entries for primes, gaps between

Crossrefs

Cf. A083371, A124589, A031924.

Programs

PARI

v=List([2]);p=3;forprime(q=5,1e3,if(q-p<=6,listput(v,p));p=q);Vec(v) \\ Charles R Greathouse IV, Jul 31 2013

PARI

list(lim)=my(v=List(),p=2); forprime(q=3,nextprime(lim\1+1), if(q-p<7, listput(v,p)); p=q); Vec(v) \\ Charles R Greathouse IV, Jan 31 2017

Formula

A000040 MINUS A083371. - R. J. Mathar, Jun 15 2008

A124589 UNION A031924. - R. J. Mathar, Jan 23 2022

a(n) >> n log^2 n. - Charles R Greathouse IV, Jan 31 2017