OFFSET
1,1
COMMENTS
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}.
This implies the conjecture in A261361.
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..2250
Zhi-Wei Sun, Problems on combinatorial properties of primes, arXiv:1402.6641 [math.NT], 2014.
EXAMPLE
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.
MATHEMATICA
f[n_]:=2*Prime[Prime[n]]+1
PQ[p_]:=PrimeQ[p]&&PrimeQ[PrimePi[p]]
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}]
CROSSREFS
KEYWORD
nonn
AUTHOR
Zhi-Wei Sun, Aug 16 2015
STATUS
approved