A260121 - OEIS

A260121

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

%I #9 Jul 17 2015 04:55:41

%S 1,1,1,2,4,8,45,1,15,34,9,146,63,128,9,20,79,45,242,50,44,71,103,181,

%T 98,208,5,180,162,299,710,10,3,388,144,427,225,121,79,25,580,230,471,

%U 46,3,1040,11,224,305,56,1163,104,93,193,55,90,88,521,898,218

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

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

%D 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 Zhi-Wei Sun, <a href="/A260121/b260121.txt">Table of n, a(n) for n = 1..100</a>

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

%t P[n_,p_]:=PrimeQ[p]&&Mod[PrimePi[p],n]==0

%t 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}]

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

%K nonn

%O 1,4

%A _Zhi-Wei Sun_, Jul 17 2015