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%I #9 Aug 16 2015 12:07:54
%S 2,21531,2,35434,11107,35175,24674,64624,127943,1981,155709,50657,
%T 74313,11479,6,1981,43405,40859,74229,2,154292,51711,29460,29011,
%U 42001,28352,2979,85836,6936,186608,3705,14402,25525,96192,6,113433,164,787,71873,3365,93169,47219,43128,184740,2,78329,13656,6936,139469,26713
%N Least positive integer k such that both k and k*n belong to the set {m>0: 2*prime(prime(m))+1 = prime(p) for some prime p}.
%C Conjecture: Let a,b,c be positive integers with gcd(a,b) = gcd(a,c) = gcd(b,c) = 1. If a+b+c is even and a is not equal to b, then any positive rational number r can be written as m/n with m and n in the set {k>0: a*prime(p) - b*prime(prime(k)) = c for some prime p}.
%C This implies the conjecture in A261361.
%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="/A261362/b261362.txt">Table of n, a(n) for n = 1..2250</a>
%H Zhi-Wei Sun, <a href="/A261362/a261362.txt">Checking the conjecture for (a,b,c) = (1,2,1) and r = s/t (s,t = 1..70)</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) = 21531 since 2*prime(prime(21531))+1 = 2*prime(243799)+1 = 2*3403703+1 = 6807407 = prime(464351) with 464351 prime, and 2*prime(prime(21531*2))+1 = 2*prime(520019)+1 = 2*7686083+1 = 15372167 = prime(993197) with 993197 prime.
%t f[n_]:=2*Prime[Prime[n]]+1
%t PQ[p_]:=PrimeQ[p]&&PrimeQ[PrimePi[p]]
%t Do[k=0;Label[bb];k=k+1;If[PQ[f[k]]&&PQ[f[k*n]],Goto[aa],Goto[bb]];Label[aa];Print[n," ", k];Continue,{n,1,50}]
%Y Cf. A000040, A005384, A260886, A261353, A261354, A261361.
%K nonn
%O 1,1
%A _Zhi-Wei Sun_, Aug 16 2015
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