# Central numbers and de Polignac's conjecture

Hello,

Let's define the notion of "central number" as follows: $n$ is said to be a central number if and only if there exist two consecutive odd primes $p$ and $q$ such that $n=(p+q)/2$. It is quite easy to check that the number of central numbers less than $x$ is asymptotically equal to $\pi(x)$. Let's now say that a central number $n$ is of type $k$ if and only if the least prime factor of $n$ equals $p_k$.

I have several questions:
1) Letting $N_k(x)$ be the number of central numbers of type $k$ less than $x$, is it true that $\displaystyle{\lim_{x\to\infty}\dfrac{N_{k}(x)}{\pi(x)}}$ exists and is positive for all positive $k$?
2) Does de Polignac's conjecture imply that there exists infinitely many central numbers of type $k$ for all positive $k$?
3) Does an affirmative answer to question 1) "conversely" imply de Polignac's conjecture?