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%I #11 Jul 15 2015 20:38:59
%S 5,18,18,9,115,208,69,373,68,430,8,214,57,1887,1255,295,880,542,5612,
%T 767,1562,40,853,884,753,4332,4750,6077,799,1394,639,5442,4785,440,
%U 7417,1290,15830,27745,3927,5701,1891,22008,8243,6031,9172,5949,43286,20778,9876,12472
%N Least positive integer k such that prime(k*n)^2 - 2 = prime(i*n)*prime(j*n) for some integers 0 < i < j.
%C Conjecture: a(n) exists for any n > 0.
%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="/A260080/b260080.txt">Table of n, a(n) for n = 1..105</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(1) = 5 since prime(5*1)^2-2 = 11^2-2 = 119 = 7*17 = prime(4*1)*prime(7*1).
%e a(66) = 149073 since prime(149073*66)^2-2 = 176365951^2-2 = 31104948672134399 = 3160879*9840600881 = prime(3448*66)*prime(9840600881*66).
%t Dv[n_]:=Divisors[Prime[n]^2-2]
%t L[n_]:=Length[Dv[n]]
%t P[k_,n_]:=L[k*n]==4&&PrimeQ[Part[Dv[k*n],2]]&&Mod[PrimePi[Part[Dv[k*n],2]],n]==0&&PrimeQ[Part[Dv[k*n],3]]&&Mod[PrimePi[Part[Dv[k*n],3]],n]==0
%t Do[k=0;Label[bb];k=k+1;If[P[k,n],Goto[aa]];Goto[bb];Label[aa];Print[n," ", k];Continue,{n,1,50}]
%Y Cf. A000040, A062326, A253257, A257926, A257938, A260078.
%K nonn
%O 1,1
%A _Zhi-Wei Sun_, Jul 15 2015
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