Given a strictly increasing sequence $S$ of positive integers $S(1),\cdots,S(n)$, define the arithmetic complexity $k(S)$ of $S$ as the infimum of the integers $k$ such that $S$ can be partitioned into $k$ disjoint arithmetic progressions. Say $S$ is weakly almost arithmetic if $k(S)$ is an $o(n)$ as $n$ tends to infinity, strongly almost arithmetic if for any $\varepsilon>0$, $k(S)\ll_{\varepsilon}n^{\varepsilon}$. Further define the arithmeticity coefficient of $S$ as $1-\log(k(S))/\log n$, so that an arithmetic sequence, for which $k(S)=1$ is 100 percent arithmetic.
Let's now consider a composite integer $m$ and the sequence of the positive integers $r_{i}(m)$ such that for all $i$, both $n-r_{i}(m)$ and $n+r_{i}(m)$ are prime and $i<j$ implies $r_{i}(m)\lt r_{j}(m)$. Is the sequence of the $r_{i}(m)$ weakly (resp. strongly) almost arithmetic? If only the answer to the first question is yes, what is the arithmeticity coefficient of the sequence of the $r_{i}(m)$ as both $m$ and the number of the $r_{i}(m)$ tend to infinity?