Counting Residue-Class Survivor Sets: An Asymptotic Comparison with Prime-Admissible Sets

For $n\geq1$, choose one residue class $r_k\pmod{k}$ for every $2\leq k\leq n$, and retain those $m\in\{2,\ldots,n+1\}$ for which $m\not\equiv r_k\pmod{k}$ whenever $k<m$. Let $N(n)$ be the number of distinct survivor sets obtained in this way. We prove \[ \log 2\leq \liminf_{n\to\infty}\frac{\log N