# Prime gap assuming Goldbach's conjecture

Assuming Goldbach's conjecture, let's denote by $r_{0}(n)$ the smallest primality radius of $n$ (non negative integer $r$ such that both $n-r$ and $n+r$ are primes) and by $r_{i+1}(n)$ the smallest primality radius of $n$ greater than $r_{i}(n)$. Let's denote by $k_{i}(n)$ the quantity $\pi(n+r_{i}(n))-\pi(n-r_{i}(n))$ where $\pi(x)$ is the number of primes not exceeding $x$.

Is it true that there exists $C>0$ such that

$\qquad \qquad (1/C)k_{i}(n)/i<2r_{i}(n)/k_{i}(n)<C.k_{i}(n)/i \qquad$?

If yes, can one take $C<2$?

$\qquad n=24 \\ \qquad \begin{array} {l}\hline r_{0}(n)=5 & k_{0}(n)=2 \\ r_{1}(n)=7 & k_{1}(n)=4 & 2×7/2=3.5 & 4/1=4 \\ r_{2}(n)=13& k_{2}(n)=7 & 2×13/7=3.72 & 7/2=3.5 \\ \end{array}$
$\qquad n=35 \\ \qquad \begin{array} {l}\hline r_{0}(n)=6 & k_{0}(n)=3 \\ r_{1}(n)=12 & k_{1}(n)=6& 2×12/6=4& 6/1=6 \\ r_{2}(n)=18& k_{2}(n)=9& 2×18/9=4& 9/2=4.5 \end{array}$