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

%I A257663 #12 Mar 12 2019 17:57:41
%S A257663 2,24,57,32,388,37,15,28,97,67,112,137,654,8,37,33,1092,1479,3390,15,
%T A257663 77,63,3,676,36,183,9,2,252,341,5,17,3,260,2,7,193,166,7,3,1102,7,297,
%U A257663 122,2,807,387,493,41,1029,189,746,79,28850,467,4,93,559,2026
%N A257663 Least positive integer k such that prime(k*n) - prime(k+n) is a square.
%C A257663 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.
%D A257663 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 A257663 Zhi-Wei Sun, <a href="/A257663/b257663.txt">Table of n, a(n) for n = 2..600</a>
%H A257663 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 A257663 a(2) = 2 since prime(2*2) - prime(2+2) = 7 - 7 = 0^2.
%e A257663 a(3) = 24 since prime(24*3) - prime(24+3) = 359 - 103 = 16^2.
%t A257663 SQ[n_]:=IntegerQ[Sqrt[n]]
%t A257663 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}]
%t A257663 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 *)
%Y A257663 Cf. A000040, A000290, A257856, A259712.
%K A257663 nonn
%O A257663 2,1
%A A257663 _Zhi-Wei Sun_, Jul 12 2015