A257663 Least positive integer k such that prime(k*n) - prime(k+n) is a square.

2, 24, 57, 32, 388, 37, 15, 28, 97, 67, 112, 137, 654, 8, 37, 33, 1092, 1479, 3390, 15, 77, 63, 3, 676, 36, 183, 9, 2, 252, 341, 5, 17, 3, 260, 2, 7, 193, 166, 7, 3, 1102, 7, 297, 122, 2, 807, 387, 493, 41, 1029, 189, 746, 79, 28850, 467, 4, 93, 559, 2026

Offset: 2

Author

Zhi-Wei Sun, Jul 12 2015

Keywords

• nonn

Comments

Conjecture: a(n) exists for any n > 1. Also, for any n > 0 there is a number k > 0 such that prime(k*n) + prime(k+n) is a square.

Examples

			a(2) = 2 since prime(2*2) - prime(2+2) = 7 - 7 = 0^2.
a(3) = 24 since prime(24*3) - prime(24+3) = 359 - 103 = 16^2.
		

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 = 2..600
• Zhi-Wei Sun, Problems on combinatorial properties of primes, arXiv:1402.6641 [math.NT], 2014.

Crossrefs

Cf. A000040, A000290, A257856, A259712.

Programs

Mathematica

SQ[n_]:=IntegerQ[Sqrt[n]]
Do[k=0; Label[bb]; k=k+1; If[SQ[Prime[k*n]-Prime[k+n]], Goto[aa], Goto[bb]]; Label[aa]; Print[n, " ", k]; Continue,{n,2,60}]
lpi[n_]:=Module[{k=1},While[!IntegerQ[Sqrt[Prime[k*n]-Prime[k+n]]], k++]; k]; Array[lpi,60,2] (* Harvey P. Dale, Mar 12 2019 *)