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%I #11 Aug 15 2015 02:04:42
%S 14,60,135,41,199,2,2,2,61,2,183,25,15,12,47,143,110,294,117,88,22,
%T 402,26,269,116,145,164,6,10,488,2,44,120,4,127,144,119,704,1058,368,
%U 104,2,6,214,4,129,2,3,301,2,2,466,20,107,280,14,337,12,22,12,242,1705,415,10,115,50,2,420,4,15
%N Least positive integer k such that prime(k)*prime(k*n) = prime(p)+2 for some prime p.
%C Conjecture: a(n) exists for any n > 0. In general, any positive rational number r can be written as m/n, where m and n are positive integers such that prime(m)*prime(n) = prime(p)+2 for some prime p.
%C For example, 14/19 = 24528/33288, and prime(24528)*prime(33288) = 281153*392723 = 110415249619 = prime(4528436431)+2 with 4528436431 prime.
%C The conjecture implies that there are infinitely many primes p such that prime(p)+2 is a product of two primes. Recall that a prime p is called a Chen prime if p+2 is a product of at most two primes.
%D Jing-run Chen, On the representation of a large even integer as the sum of a prime and a product of at most two primes, Sci. Sinica 16(1973), 157-176.
%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="/A261282/b261282.txt">Table of n, a(n) for n = 1..2000</a>
%H Zhi-Wei Sun, <a href="/A261282/a261282.txt">Checking the conjecture for r = a/b with a,b = 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(2) = 60 since prime(60)*prime(60*2) = 281*659 = 185179 = prime(16763)+2 with 16763 prime.
%t f[n_]:=Prime[n]
%t PQ[n_]:=PrimeQ[n]&&PrimeQ[PrimePi[n]]
%t Do[k=0;Label[bb];k=k+1;If[PQ[f[k]*f[k*n]-2],Goto[aa],Goto[bb]];Label[aa];Print[n," ", k];Continue,{n,1,70}]
%Y Cf. A000040, A109611, A257926, A259487, A261281.
%K nonn
%O 1,1
%A _Zhi-Wei Sun_, Aug 14 2015