# Bounds for sets of remainders

## TL;DR

This paper studies the sequence counting distinct remainders for numbers modulo all smaller divisors, providing asymptotic bounds, analyzing differences between terms, and exploring related iterated remainder sets.

## Contribution

It establishes explicit asymptotic formulas for the sequence, bounds on term differences, and analyzes properties of iterated remainder sets related to Pierce expansions.

## Key findings

- s(n) = c * n + O(n/(\log n \log \log n))
- Differences between s(n) and s(n+1) are at most one, with arbitrarily large decreases
- Bounds are provided for the size of iterated remainder sets

## Abstract

Let $s(n)$ be the number of different remainders $n \bmod k$, where $1 \leq k \leq \lfloor n/2 \rfloor$. This rather natural sequence is sequence A283190 in the OEIS and while some basic facts are known, it seems that surprisingly it has barely been studied. First, we prove that $s(n) = c \cdot n + O(n/(\log n \log \log n))$, where $c$ is an explicit constant. Then we focus on differences between consecutive terms $s(n)$ and $s(n+1)$. It turns out that the value can always increase by at most one, but there exist arbitrarily large decreases. We show that the differences are bounded by $O(\log \log n)$. Finally, we consider ''iterated remainder sets''. These are related to a problem arising from Pierce expansions, and we prove bounds for the size of these sets as well.