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%I #30 Sep 11 2018 03:59:13
%S 3,5,11,157,1536160091,9918821194603,341976204789992332577,
%T 2166703103992332274919569
%N 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.
%C Second term in the smallest non-constant p-term arithmetic progression (AP) of primes beginning with p = prime(n).
%C 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.
%C 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."
%C Phil Carmody found a(7) in 2001.
%C See the crossrefs for more comments, references, and links.
%D P. Ribenboim, My Numbers, My Friends, Springer, 2000; p. 67.
%D P. Ribenboim, The Book of Prime Number Records, 2nd ed., Springer, 1989; p. 225.
%H Phil Carmody, <a href="https://listserv.nodak.edu/cgi-bin/wa.exe?A2=NMBRTHRY;8b742242.0111">a(7)</a>, NMBRTHRY November 2001.
%H <a href="/index/Pri#primes_AP">Index entries for sequences related to primes in arithmetic progressions</a>
%F a(n) = prime(n) + A088430(n) = prime(n) + A002110(n)*A231018(n).
%e Prime(3) = 5 and 5, 11, 17, 23, 29 is the smallest 5-term AP beginning with 5, so a(3) = 11.
%o (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
%Y For common differences see A088430, for initial terms see A000040, for last terms see A113834, for the APs see A231406.
%Y For other types of APs of primes see A005115 and its comments.
%K hard,more,nonn
%O 1,1
%A _Jonathan Sondow_, Nov 08 2013
%E a(8) found by Wojciech Izykowski, from _Jens Kruse Andersen_, Jun 30 2014
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