cp's OEIS Frontend

A261437 Least positive integer k such that k*n+1 = prime(p) and k^2*n+1 = prime(q) for some pair of primes p and q.

%I A261437 #22 Aug 18 2015 11:17:14
%S A261437 2,1,286,1,7290,21,18,2472,12,1,20460,20,20692,105,4392,1,96816,1327,
%T A261437 360,264,19850,2734,1854,5293,930,29526,98,622,9222,1,6816,924,61614,
%U A261437 70,53760,45,32190,9687,5510,1,128070,148,8772,23478,404,801,1830,5,9912,7662,1100,8211,1116,9997,630,4965,936,1,87570,759
%N A261437 Least positive integer k such that k*n+1 = prime(p) and k^2*n+1 = prime(q) for some pair of primes p and q.
%C A261437 Conjecture: (i) If n > 0 and r are relatively prime integers, then there are infinitely many positive integers k such that k*n+r = prime(p) for some prime p.
%C A261437 (ii) Let r be 1 or -1. For any integer n > 0, there is a positive integer k such that k*n+r = prime(p) and k^2*n+1 = prime(q) for some primes p and q.
%C A261437 (iii) For any integer n > 0, there is a positive integer k such that n+k = prime(p) and n+k^2 = prime(q) for some primes p and q.
%C A261437 Note that part (i) is a refinement of Dirichlet's theorem on primes in arithmetic progressions. Also, part (ii) implies that a(n) exists for any n > 0.
%D A261437 Zhi-Wei Sun, Problems on combinatorial properties of primes, in: M. Kaneko, S. Kanemitsu and J. Liu (eds.), Number Theory: Plowing and Starring through High Wave Forms, Proc. 7th China-Japan Seminar (Fukuoka, Oct. 28 - Nov. 1, 2013), Ser. Number Theory Appl., Vol. 11, World Sci., Singapore, 2015, pp. 169-187.
%H A261437 Zhi-Wei Sun, <a href="/A261437/b261437.txt">Table of n, a(n) for n = 1..200</a>
%H A261437 Zhi-Wei Sun, <a href="http://arxiv.org/abs/1402.6641">Problems on combinatorial properties of primes</a>, arXiv:1402.6641 [math.NT], 2014.
%e A261437 a(3) = 286 since 286*3+1 = 859 = prime(149) with 149 prime, and 286^2*3+1 = 245389 = prime(21661) with 21661 prime.
%t A261437 PQ[p_]:=PrimeQ[p]&&PrimeQ[PrimePi[p]]
%t A261437 Do[k=0;Label[bb];k=k+1;If[PQ[k*n+1]&&PQ[k^2*n+1],Goto[aa],Goto[bb]];Label[aa];Print[n," ", k];Continue,{n,1,60}]
%Y A261437 Cf. A000040, A000290, A034693, A053989, A185636, A204065.
%K A261437 nonn
%O A261437 1,1
%A A261437 _Zhi-Wei Sun_, Aug 18 2015