A260121 Least positive integer k such that prime(k*n)^2 - 2 = prime(j*n) for some j > 0.

1, 1, 1, 2, 4, 8, 45, 1, 15, 34, 9, 146, 63, 128, 9, 20, 79, 45, 242, 50, 44, 71, 103, 181, 98, 208, 5, 180, 162, 299, 710, 10, 3, 388, 144, 427, 225, 121, 79, 25, 580, 230, 471, 46, 3, 1040, 11, 224, 305, 56, 1163, 104, 93, 193, 55, 90, 88, 521, 898, 218

Offset: 1

Author

Zhi-Wei Sun, Jul 17 2015

Keywords

• nonn

Comments

The conjecture in A260120 implies that a(n) exists for any n > 0, which is stronger than the conjecture in A253257.

Examples

			a(5) = 4 since prime(4*5)^2-2 = 71^2-2 = 5039 = prime(135*5).
		

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..100
• Zhi-Wei Sun, Problems on combinatorial properties of primes, arXiv:1402.6641 [math.NT], 2014.

Crossrefs

Cf. A000040, A062326, A253257, A260080, A260120.

Programs

Mathematica

P[n_,p_]:=PrimeQ[p]&&Mod[PrimePi[p],n]==0
Do[k=0;Label[bb];k=k+1; If[P[n,Prime[k*n]^2-2],Goto[aa]];Goto[bb];Label[aa];Print[n, " ", k];Continue,{n,1,60}]