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

TL;DR

This paper characterizes positive integers N for which the sets of small and large divisors, excluding trivial divisors, satisfy linear recurrences of order at most two, extending previous results and increasing analysis complexity.

Contribution

It extends prior characterizations of divisor sets satisfying linear recurrences by excluding trivial divisors, providing new classifications for both small and large divisor sets.

Findings

Characterization of N with small divisor sets satisfying linear recurrence

Characterization of N with large divisor sets satisfying linear recurrence

Extension of previous results on divisor sets in arithmetic progression

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 $S'_N$ 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.