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

Abstract

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 $S_N^{\prime }$ satisfies a linear recurrence of order at most two. Moreover, our second result characterizes all positive $N$ such that $L_N^{\prime }$ satisfies a linear recurrence of order at most two, thus extending considerably a recent result that characterizes $N$ with $L_N^{\prime }$ being in an arithmetic progression.