On numbers $n$ dividing the $n$th term of a linear recurrence

TL;DR

This paper establishes bounds on how many positive integers up to a certain limit divide the corresponding term in a nondegenerate linear recurrence sequence with simple roots.

Contribution

It provides new upper and lower bounds on the count of such integers, advancing understanding of divisibility properties in linear recurrence sequences.

Findings

Derived explicit upper bounds for the count of dividing integers.

Established lower bounds indicating the frequency of such divisors.

Analyzed sequences with simple roots to generalize previous results.

Abstract

Here, we give upper and lower bounds on the count of positive integers $n\le x$ dividing the $n$th term of a nondegenerate linearly recurrent sequence with simple roots.