

A231017


Least prime q > p = prime(n) such that if d = qp, then p, p+d, p+2d, ..., p+(p1)d are all primes.


3




OFFSET

1,1


COMMENTS

Second term in the smallest nonconstant pterm arithmetic progression (AP) of primes beginning with p = prime(n).
For any nonconstant 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.


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

Table of n, a(n) for n=1..8.
Phil Carmody, a(7), NMBRTHRY November 2001.
Index entries for sequences related to primes in arithmetic progressions


FORMULA

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


EXAMPLE

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


PROG

(PARI) a(n)=my(p=prime(n), P=prod(i=1, n1, prime(i)), d); forprime(q=p+1, , d=qp; if(d%P, next); for(i=2, p1, if(!isprime(p+i*d), next(2))); return(q)) \\ Charles R Greathouse IV, Nov 08 2013


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.
Sequence in context: A277552 A154941 A062601 * A038198 A280876 A079037
Adjacent sequences: A231014 A231015 A231016 * A231018 A231019 A231020


KEYWORD

hard,more,nonn


AUTHOR

Jonathan Sondow, Nov 08 2013


EXTENSIONS

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


STATUS

approved



