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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 $m$?

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