  For each positive integer $N$, define $$S'_N \ =\ \{1 < d < \sqrt{N}:
d|N\}\mbox{ and }L'_N \ =\ \{\sqrt{N} < d < N : d|N\}.$$ Recently, Chentouf
characterized all positive integers $N$ such that the set of small divisors
$\{d\le \sqrt{N}: d|N\}$ 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 $S'_N$ satisfies a
linear recurrence of order at most two. Moreover, our second result
characterizes all positive $N$ such that $L'_N$ satisfies a linear recurrence
of order at most two, thus extending considerably a recent result that
characterizes $N$ with $L'_N$ being in an arithmetic progression.
