Linear recurrences of order at most two in nontrivial small divisors and large divisors

For each positive integer $N$, define $$S_N^{\mathrm{\prime }}=\{1<d<\sqrt{N}:d\mathrm{\mid }N\}and{L}_N^{\mathrm{\prime }}=\{\sqrt{N}<d<N:d\mathrm{\mid }N\}\mathrm{.}$$ Recently, Chentouf characterized all positive integers $N$ such that the set of small divisors $\{d\le \sqrt{N}:d\mathrm{\mid }N\}$ satisfies a linear recurrence of order at most two. We nontrivially extend the result by excluding the trivial divisor $1$ from consideration, which dramatically increases the analysis complexity. Our first result characterizes all positive integers $N$ such that $S_N^{\mathrm{\prime }}$ satisfies a linear recurrence of order at most two. Moreover, our second result characterizes all positive $N$ such that $L_N^{\mathrm{\prime }}$ satisfies a linear recurrence of order at most two, thus extending considerably a recent result that characterizes $N$ with $L_N^{\mathrm{\prime }}$ being in an arithmetic progression.