Linear Recurrences of Order at Most Two in Small Divisors

TL;DR

This paper classifies all positive integers n where the small divisors (not exceeding sqrt(n)) follow a linear recurrence of order at most two, extending previous work on arithmetic progressions.

Contribution

It generalizes Ianucci's classification from arithmetic progressions to linear recurrences of order at most two for small divisors.

Findings

Complete classification of n with small divisors forming linear recurrences of order ≤ 2.

Extension of previous classification from arithmetic progressions to linear recurrences.

Provides explicit conditions characterizing such n.

Abstract

Given a positive integer $n$, the small divisors of $n$ are defined as the positive divisors that do not exceed $\sqrt{n}.$ Ianucci previously classified all $n$ for which the small divisors of $n$ form an arithmetic progression. In this paper, we classify all $n$ for which the small divisors of $n$ form a linear recurrence of order at most two.