A259488 Positive integers k with prime(k)+2 and prime(prime(k))+2 both prime.

2, 3, 7, 13, 296, 343, 395, 405, 408, 463, 469, 473, 542, 572, 577, 584, 625, 671, 673, 695, 837, 984, 1016, 1030, 1074, 1165, 1224, 1230, 1328, 1410, 1445, 1679, 1825, 1860, 1867, 1949, 2078, 2091, 2095, 2123, 2167, 2476, 2478, 2616, 2753, 2764, 2956, 3011, 3065, 3416, 3621, 3646, 3712, 3720, 3758, 3872, 3926, 4063, 4071, 4079, 4133, 4217, 4312, 4351, 4524, 4745, 4855, 4865, 4882, 4922

Offset: 1

Author

Zhi-Wei Sun, Jun 28 2015

Keywords

• nonn

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.

Examples

			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.
		

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.

Crossrefs

Cf. A000040, A001359, A006512, A236458, A236470, A259487.

Programs

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 *)

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