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%I #29 Jul 03 2015 06:18:24
%S 2,1860,408,25011,51312,37977,695,4071,10970,3621,17671,12005,1230,
%T 19494,542,577,408,2476,584,542,469,34229,343,24078,3011,25749,20706,
%U 24198,2478,3926,1030,1030,13857,3621,343,13380,2476,4922,2476,296,19176,29175,34737,13,625,2956,408,572,7,469,15604,9699,26515,2167,5302,9773,54254,1410,4524,4351
%N Least positive integer m with prime(m)+2 and prime(prime(m))+2 both prime such that prime(m*n)+2 and prime(prime(m*n))+2 are both prime.
%C Conjecture: Any positive rational number r can be written as m/n with m and n terms of A259488.
%C This implies that there are infinitely many primes p with p+2 and prime(p)+2 both prime.
%C I have verified the conjecture for all those r = a/b with a,b = 1,...,400. - _Zhi-Wei Sun_, Jun 29 2015
%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="/A259487/b259487.txt">Table of n, a(n) for n = 1..5000</a>
%H Zhi-Wei Sun, <a href="/A259487/a259487_1.txt">Checking the conjecture for r = a/b with a,b = 1..400</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) = 2 since prime(2)+2 = 3+2 = 5 and prime(prime(2))+2 = prime(3)+2 = 7 are both prime, but prime(1)+2 = 4 is composite.
%e a(49) = 7 since prime(7)+2 = 17+2 = 19, prime(prime(7))+2 = prime(17)+2 = 59+2 = 61, prime(49*7)+2 = 2309+2 = 2311 and prime(prime(49*7))+2 = prime(2309)+2 = 20441+2 = 20443 are all prime.
%t PQ[k_]:=PrimeQ[Prime[k]+2]&&PrimeQ[Prime[Prime[k]]+2]
%t Do[k=0;Label[bb];k=k+1;If[PQ[k]&&PQ[n*k], Goto[aa], Goto[bb]];Label[aa];Print[n," ", k];Continue,{n,1,60}]
%Y Cf. A000040, A001359, A006512, A236458, A258803, A258836, A259488, A259492.
%K nonn
%O 1,1
%A _Zhi-Wei Sun_, Jun 28 2015
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