A261361 - OEIS

%I #9 Aug 16 2015 12:07:36

%S 3,13,173,463,523,823,971,991,1291,1543,2113,4003,4019,4649,5923,6037,

%T 6101,7649,10103,12539,12841,17203,17569,18013,21193,22093,23321,

%U 25111,27197,31231,32009,32117,33349,34687,35423,35449,37747,39619,41729,41759,42017,43237,43331,44741,45841,50021,51437,52489,55921,56891

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

%C Conjecture: The sequence contains infinitely many terms. In general, if a,b,c are positive integers with gcd(a,b) = gcd(a,c) = gcd(b,c) = 1, and a+b+c is even and a is not equal to b, then there are infinitely many prime pairs {p,q} such that a*prime(p) - b*prime(q) = c.

%C See also A261362 for a stronger conjecture.

%C Recall that a prime p is called a Sophie Germain prime if 2*p+1 is also prime. A well-known conjecture states that there are infinitely many Sophie Germain primes.

%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="/A261361/b261361.txt">Table of n, a(n) for n = 1..10000</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) = 3 since 3 is a prime, and 2*prime(3)+1 = 2*5+1 = 11 = prime(5) with 5 prime.

%e a(3) = 173 since 173 is a prime, and 2*prime(173)+1 = 2*1031+1 = 2063 = prime(311) with 311 prime.

%t f[n_]:=2*Prime[Prime[n]]+1

%t PQ[p_]:=PrimeQ[p]&&PrimeQ[PrimePi[p]]

%t n=0;Do[If[PQ[f[k]],n=n+1;Print[n," ",Prime[k]]],{k,1,5800}]

%Y Cf. A000040, A005384, A260886, A260888, A261352, A261354, A261362.

%K nonn

%O 1,1

%A _Zhi-Wei Sun_, Aug 16 2015