# Asymptotics for the $m$-th integer of given fundamental primality radius and order of centrality

Assuming Goldbach's conjecture, define the "fundamental primality radius of $$n$$", denoted by $$r_{0}(n)$$, as $$\inf\{r>0\mid (n-r,n+r)\in\mathbb{P}^{2}\}$$, and the "order of centrality of $$n$$", denoted by $$k_{0}(n)$$, as $$\pi(n+r_{0}(n))-\pi(n-r_{0}(n))$$. Denote also by $$\mathbb{G}(r,k):=\{n>0\mid(r_{0}(n),k_{0}(n))=(r,k)\}$$, by $$\pi_{(r,k)}(x):=\#(\mathbb{G}(r,k)\cap(1,x])$$ and finally by $$(r,k)_{m}$$ the integer equal to $$\inf\{x\mid\pi_{(r,k)}(x)=m\}$$.

Is it possible, assuming that $$\mathbb{G}(r,k)\neq\emptyset\Longrightarrow \lim_{x\to\infty}\pi_{(r,k)}(x)=\infty$$ to give an asymptotics for $$(r,k)_{m}$$ in terms of $$m$$?