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%I #7 Jul 15 2015 10:06:41
%S 2,2,2,21,9,10,12,14,47,32,32,171,177,175,64,187,330,206,77,467,4,126,
%T 127,355,279,982,249,1930,105,109,659,801,269,777,703,125,819,1347,
%U 904,1153,549,2344,757,1301,1793,303,105,3168,2645,3055,110,1619,1580,2423,220,965,1397,84,988,322
%N Least positive integer k such that (prime(k*n)-1)^2 = (prime(i*n)-1)*(prime(j*n)-1) for some integers 0 < i < j.
%C Conjecture: a(n) exists for any n > 0. In general, for any nonzero integer m and positive integer n there are distinct positive integers i,j,k such that (prime(i*n)+m)*(prime(j*n)+m) = (prime(k*n)+m)^2.
%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="/A260082/b260082.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(4) = 21 since (prime(21*4)-1)^2 = 432^2 = 18*10368 = (prime(2*4)-1)*(prime(318*4)-1).
%e a(61) = 15160 since (prime(15160*61)-1)^2 = 14242116^2 = 47316*4286876916 = (prime(80*61)-1)*(prime(3326491*61)-1).
%t Dv[n_]:=Divisors[(Prime[n]-1)^2]
%t L[n_]:=Length[Dv[n]]
%t P[k_,n_,i_]:=PrimeQ[Part[Dv[k*n],i]+1]&&Mod[PrimePi[Part[Dv[k*n],i]+1],n]==0
%t Do[k=0;Label[bb];k=k+1; Do[If[P[k,n,i]&&P[k,n,L[k*n]-i+1],Goto[aa]],{i,1,L[k*n]/2}];Goto[bb];Label[aa];Print[n, " ", k];Continue,{n,1,60}]
%Y Cf. A000040, A257926, A257928, A257938, A260078, A260080.
%K nonn
%O 1,1
%A _Zhi-Wei Sun_, Jul 15 2015
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