A238580 - OEIS

%I #8 Mar 01 2014 10:11:08

%S 1,1,2,1,2,2,1,1,3,1,2,3,2,3,3,3,2,4,2,3,3,3,2,3,2,2,3,1,1,5,3,4,3,1,

%T 4,3,1,5,4,4,2,4,5,4,5,2,5,5,3,2,4,2,4,5,3,5,2,7,4,5,2,5,4,8,4,6,5,6,

%U 5,2,5,4,3,6,2,5,1,5,8,4

%N a(n) = |{0 < k <= n: 2*k + 1 and prime(k)*prime(n) - 2 are both prime}|.

%C Conjecture: a(n) > 0 for all n > 0, and a(n) = 1 only for n = 1, 2, 4, 7, 8, 10, 28, 34,37, 77.

%C Note that a prime p with p + 2 a product of at most two primes is called a Chen prime.

%D J.-R. 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.

%H Zhi-Wei Sun, <a href="/A238580/b238580.txt">Table of n, a(n) for n = 1..10000</a>

%H Zhi-Wei Sun, <a href="http://arxiv.org/abs/1402.6641">Problems on combinatorial properties of primes</a>, arXiv:1402.6641, 2014.

%e a(7) = 1 since 2*3 + 1 = 7 and prime(3)*prime(7) - 2 = 5*17 - 2 = 83 are both prime.

%e a(8) = 1 since 2*8 + 1 = 17 and prime(8)*prime(8) - 2 = 19^2 - 2 = 359 are both prime.

%e a(77) = 1 since 2*20 + 1 = 41 and prime(20)*prime(77) - 2 = 71*389 - 2 = 27617 are both prime.

%t p[n_,k_]:=PrimeQ[2k+1]&&PrimeQ[Prime[n]*Prime[k]-2]

%t a[n_]:=Sum[If[p[n,k],1,0],{k,1,n}]

%t Table[a[n],{n,1,80}]

%Y Cf. A000040, A109611, A233529.

%K nonn

%O 1,3

%A _Zhi-Wei Sun_, Mar 01 2014