A231017 Least prime q > p = prime(n) such that if d = q-p, then p, p+d, p+2d, ..., p+(p-1)d are all primes.

3, 5, 11, 157, 1536160091, 9918821194603, 341976204789992332577, 2166703103992332274919569

Offset: 1

Author

Jonathan Sondow, Nov 08 2013

Keywords

• hard
• more
• nonn

Comments

Second term in the smallest non-constant p-term arithmetic progression (AP) of primes beginning with p = prime(n).

For any non-constant AP beginning with a prime p and ending with a prime, the maximum possible length is p, since p+pd is not prime for d > 0. If all the terms are prime, then the common difference d must be a multiple of all primes < p.

Ribenboim says that in 1986 G. Loh [Loeh] discovered a(5) and a(6), and that a(n) should exist for all n, but "in my opinion, this is so difficult that no one will prove [it], and no one will find a counterexample in the near future."

Phil Carmody found a(7) in 2001.

See the crossrefs for more comments, references, and links.

Examples

			Prime(3) = 5 and 5, 11, 17, 23, 29 is the smallest 5-term AP beginning with 5, so a(3) = 11.
		

References

• P. Ribenboim, My Numbers, My Friends, Springer, 2000; p. 67.
• P. Ribenboim, The Book of Prime Number Records, 2nd ed., Springer, 1989; p. 225.

Links

• Phil Carmody, a(7), NMBRTHRY November 2001.
• Index entries for sequences related to primes in arithmetic progressions

Crossrefs

For common differences see A088430, for initial terms see A000040, for last terms see A113834, for the APs see A231406.

For other types of APs of primes see A005115 and its comments.

Programs

PARI

a(n)=my(p=prime(n),P=prod(i=1,n-1,prime(i)),d); forprime(q=p+1,, d=q-p; if(d%P,next); for(i=2,p-1,if(!isprime(p+i*d), next(2))); return(q)) \\ Charles R Greathouse IV, Nov 08 2013

Formula

a(n) = prime(n) + A088430(n) = prime(n) + A002110(n)*A231018(n).

Extensions

a(8) found by Wojciech Izykowski, from Jens Kruse Andersen, Jun 30 2014