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Least prime p such that 2*prime(p*n)+1 = prime(q*n) for some prime q.
2

%I #11 Aug 02 2015 22:26:10

%S 3,47,3,13,797,89,2269,733,7877,53,14683,16267,17167,59951,10067,761,

%T 94463,12437,124561,71881,52009,6791,10061,47287,10789,19009,4813,

%U 23173,27427,18701,23011,44917,17,70937,883,727,99079,10531,18749,126541,18121,34807,29873,159473,853,165317,80627,159721,8263,411707

%N Least prime p such that 2*prime(p*n)+1 = prime(q*n) for some prime q.

%C Conjecture: a(n) exists for any n > 0. In general, if a > 1 and b are integers with a+b odd and gcd(a,b)=1, then for any positive integer n there are primes p and q such that a*prime(p*n)+b = prime(q*n).

%C This is a supplement to the conjecture in A260120. It implies that there are infinitely many Sophie Germain primes.

%H Zhi-Wei Sun, <a href="/A260882/b260882.txt">Table of n, a(n) for n = 1..200</a>

%H Zhi-Wei Sun, <a href="http://arxiv.org/abs/1211.1588">Conjectures involving primes and quadratic forms</a>, arXiv:1211.1588 [math.NT], 2012-2015.

%e a(2) = 47 since 2*prime(47*2)+1 = 2*491+1 = 983 = prime(83*2) with 47 and 83 both prime.

%e a(199) = 2784167 since 2*prime(2784167*199)+1 = 2*12290086499+1 = 24580172999 = prime(5399231*199) with 2784167 and 5399231 both prime.

%t f[n_]:=Prime[n]

%t PQ[p_,n_]:=PrimeQ[p]&&PrimeQ[PrimePi[p]/n]

%t Do[k=0;Label[bb];k=k+1;If[PQ[2*f[n*f[k]]+1,n],Goto[aa],Goto[bb]];Label[aa];Print[n," ", f[k]];Continue,{n,1,50}]

%Y Cf. A000040, A005384, A260120, A260252.

%K nonn

%O 1,1

%A _Zhi-Wei Sun_, Aug 02 2015