OFFSET
1,1
COMMENTS
Conjecture: Any positive rational number r can be written as m/n with m and n terms of A259488.
This implies that there are infinitely many primes p with p+2 and prime(p)+2 both prime.
I have verified the conjecture for all those r = a/b with a,b = 1,...,400. - Zhi-Wei Sun, Jun 29 2015
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..5000
Zhi-Wei Sun, Problems on combinatorial properties of primes, arXiv:1402.6641 [math.NT], 2014.
EXAMPLE
a(1) = 2 since prime(2)+2 = 3+2 = 5 and prime(prime(2))+2 = prime(3)+2 = 7 are both prime, but prime(1)+2 = 4 is composite.
a(49) = 7 since prime(7)+2 = 17+2 = 19, prime(prime(7))+2 = prime(17)+2 = 59+2 = 61, prime(49*7)+2 = 2309+2 = 2311 and prime(prime(49*7))+2 = prime(2309)+2 = 20441+2 = 20443 are all prime.
MATHEMATICA
PQ[k_]:=PrimeQ[Prime[k]+2]&&PrimeQ[Prime[Prime[k]]+2]
Do[k=0; Label[bb]; k=k+1; If[PQ[k]&&PQ[n*k], Goto[aa], Goto[bb]]; Label[aa]; Print[n, " ", k]; Continue, {n, 1, 60}]
CROSSREFS
KEYWORD
nonn
AUTHOR
Zhi-Wei Sun, Jun 28 2015
STATUS
approved