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.

2, 1, 286, 1, 7290, 21, 18, 2472, 12, 1, 20460, 20, 20692, 105, 4392, 1, 96816, 1327, 360, 264, 19850, 2734, 1854, 5293, 930, 29526, 98, 622, 9222, 1, 6816, 924, 61614, 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

Offset: 1

Author

Zhi-Wei Sun, Aug 18 2015

Keywords

• nonn

Comments

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.

(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.

(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.

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.

Examples

			a(3) = 286 since 286*3+1 = 859 = prime(149) with 149 prime, and 286^2*3+1 = 245389 = prime(21661) with 21661 prime.
		

References

• 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.

Links

• Zhi-Wei Sun, Table of n, a(n) for n = 1..200
• Zhi-Wei Sun, Problems on combinatorial properties of primes, arXiv:1402.6641 [math.NT], 2014.

Crossrefs

Cf. A000040, A000290, A034693, A053989, A185636, A204065.

Programs

Mathematica

PQ[p_]:=PrimeQ[p]&&PrimeQ[PrimePi[p]]
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}]