Let $n$ be a large enough composite integer and assume Goldbach's conjecture. Denote the smallest $r>0$ such that both $n-r$ and $n+r$ are prime by $r_{0}(n)$, and by $k_{0}(n):=\pi(n+r_{0}(n))-\pi(n-r_{0}(n))$.
Does one have $r_{0}(n)=\frac{k_{0}(n)}{2}\left(f_{+}(n)+f_{-}(n)\right)+O(1)$, where $\displaystyle{f_{+}(n)=\sum_{n-r_{0}(n)\leq k\leq n+r_{0}(n)}\frac{\pi(k)\Lambda(k)}{k\log k}}$ and $f_{-}(n)$ is defined similarly but with strict inequalities?