OFFSET
1,1
COMMENTS
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.
See also A261362 for a stronger conjecture.
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.
REFERENCES
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.
LINKS
Zhi-Wei Sun, Table of n, a(n) for n = 1..10000
Zhi-Wei Sun, Problems on combinatorial properties of primes, arXiv:1402.6641 [math.NT], 2014.
EXAMPLE
a(1) = 3 since 3 is a prime, and 2*prime(3)+1 = 2*5+1 = 11 = prime(5) with 5 prime.
a(3) = 173 since 173 is a prime, and 2*prime(173)+1 = 2*1031+1 = 2063 = prime(311) with 311 prime.
MATHEMATICA
f[n_]:=2*Prime[Prime[n]]+1
PQ[p_]:=PrimeQ[p]&&PrimeQ[PrimePi[p]]
n=0; Do[If[PQ[f[k]], n=n+1; Print[n, " ", Prime[k]]], {k, 1, 5800}]
CROSSREFS
KEYWORD
nonn
AUTHOR
Zhi-Wei Sun, Aug 16 2015
STATUS
approved