cp's OEIS Frontend

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

%I A261352 #12 Aug 16 2015 12:06:29
%S A261352 11,23,167,197,223,317,359,461,593,619,859,1283,1289,1327,1487,1759,
%T A261352 1879,2557,2579,2749,2879,3617,4159,4783,5081,5333,5531,5689,5783,
%U A261352 5867,6427,6521,7589,7681,7727,7753,9041,9157,9283,9479,10111,10289,10853,11261,11779,11867,12541,13309,13399,13687
%N A261352 Primes p such that prime(p)+2 = prime(q)*prime(r) for distinct primes q and r.
%C A261352 Conjecture: The sequence has infinitely many terms.
%C A261352 See also A261354 for a similar conjecture, and A261353 for a stronger conjecture.
%C A261352 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.
%D A261352 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.
%D A261352 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.
%H A261352 Zhi-Wei Sun, <a href="/A261352/b261352.txt">Table of n, a(n) for n = 1..10000</a>
%H A261352 Zhi-Wei Sun, <a href="http://arxiv.org/abs/1402.6641">Problems on combinatorial properties of primes</a>, arXiv:1402.6641 [math.NT], 2014.
%e A261352 a(1) = 11 since 11 is a prime, and prime(11)+2 = 3*11 = prime(2)*prime(5) with 2 and 5 both prime.
%e A261352 a(2) = 23 since 23 is a prime, and prime(23)+2 = 5*17 = prime(3)*prime(7) with 3 and 7 both prime.
%t A261352 Dv[n_]:=Divisors[n]
%t A261352 PQ[n_]:=PrimeQ[n]&&PrimeQ[PrimePi[n]]
%t A261352 q[n_]:=Length[Dv[n]]==4&&PQ[Part[Dv[n],2]]&&PQ[Part[Dv[n],3]]
%t A261352 f[k_]:=Prime[Prime[k]]+2
%t A261352 n=0;Do[If[q[f[k]],n=n+1;Print[n," ",Prime[k]]],{k,1,1620}]
%Y A261352 Cf. A000040, A109611, A261282, A261353, A261354, A261361.
%K A261352 nonn
%O A261352 1,1
%A A261352 _Zhi-Wei Sun_, Aug 15 2015