A261352 Primes p such that prime(p)+2 = prime(q)*prime(r) for distinct primes q and r.

11, 23, 167, 197, 223, 317, 359, 461, 593, 619, 859, 1283, 1289, 1327, 1487, 1759, 1879, 2557, 2579, 2749, 2879, 3617, 4159, 4783, 5081, 5333, 5531, 5689, 5783, 5867, 6427, 6521, 7589, 7681, 7727, 7753, 9041, 9157, 9283, 9479, 10111, 10289, 10853, 11261, 11779, 11867, 12541, 13309, 13399, 13687

Offset: 1

Author

Zhi-Wei Sun, Aug 15 2015

Keywords

• nonn

Comments

Conjecture: The sequence has infinitely many terms.

See also A261354 for a similar conjecture, and A261353 for a stronger conjecture.

Recall that a prime p is called a Chen prime if p+2 is a product of at most two primes. It is known that there are infinitely many Chen primes.

Examples

			a(1) = 11 since 11 is a prime, and prime(11)+2 = 3*11 = prime(2)*prime(5) with 2 and 5 both prime.
a(2) = 23 since 23 is a prime, and prime(23)+2 = 5*17 = prime(3)*prime(7) with 3 and 7 both prime.
		

References

• Jing-run Chen, On the representation of a large even integer as the sum of a prime and a product of at most two primes, Sci. Sinica 16(1973), 157-176.
• 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, A109611, A261282, A261353, A261354, A261361.

Programs

Mathematica

Dv[n_]:=Divisors[n]
PQ[n_]:=PrimeQ[n]&&PrimeQ[PrimePi[n]]
q[n_]:=Length[Dv[n]]==4&&PQ[Part[Dv[n],2]]&&PQ[Part[Dv[n],3]]
f[k_]:=Prime[Prime[k]]+2
n=0;Do[If[q[f[k]],n=n+1;Print[n," ",Prime[k]]],{k,1,1620}]