OFFSET
1,1
COMMENTS
The conjecture in A259487 essentially says that {a(m)/a(n): m,n = 1,2,3,...} coincides with the set of all positive rational numbers. This implies that the current sequence has infinitely many terms.
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) = 2 since prime(2)+2 = 5 and prime(prime(2))+2 = prime(3)+2 = 7 are both prime, but prime(1)+2 = 4 is composite.
a(2) = 3 since prime(3)+2 = 7 and prime(prime(3))+2 = prime(7)+2 = 19 are both prime.
MATHEMATICA
n=0; Do[If[PrimeQ[Prime[k]+2]&&PrimeQ[Prime[Prime[k]]+2], n=n+1; Print[n, " ", k]], {k, 1, 5000}]
Select[Range[5000], AllTrue[{Prime[#]+2, Prime[Prime[#]]+2}, PrimeQ]&] (* The program uses the AllTrue function from Mathematica version 10 *) (* Harvey P. Dale, Feb 18 2018 *)
PROG
(PARI) k=pk=0; forprime(ppk=2, 1e6, if(isprime(pk++), k++; if(isprime(pk+2) && isprime(ppk+2), print1(k", ")))) \\ Charles R Greathouse IV, Jun 29 2015
CROSSREFS
KEYWORD
nonn
AUTHOR
Zhi-Wei Sun, Jun 28 2015
STATUS
approved