Counting Survivor Sets: Exponential Equivalence with Prime-Admissible Sets

For each integer $n\geq 1$, let $N(n)$ denote the number of distinct subsets of $\{2,\ldots,n+1\}$ obtained by choosing one forbidden residue class modulo each integer from $2$ to $n$; this is OEIS sequence A396595 (https://oeis.org/A396595). Equivalently, $N(n)$ is the initial-restriction complexit