# Combinatorial properties of sparsely totient numbers

## TL;DR

This paper investigates the properties and distribution of sparsely totient numbers, revealing their divisibility patterns, constructing infinite families, and analyzing their additive and multiplicative structures.

## Contribution

It provides new results on divisibility, explicit constructions, and the combinatorial structure of sparsely totient numbers, connecting them with prime distribution and progressions.

## Key findings

- Squarefree integers divide all large sparsely totient numbers
- Constructed infinite families of sparsely totient numbers
- Proved sparsely totient numbers are multiplicatively piecewise syndetic

## Abstract

Let $N_1(m)=\max\{n \colon \phi(n) \leq m\}$ and $N_1 = \{N_1(m) \colon m \in \phi(\mathbb{N})\}$ where $\phi(n)$ denotes the Euler's totient function. Masser and Shiu \cite{masser} call the elements of $N_1$ as `sparsely totient numbers' and initiated the study of these numbers. In this article, we establish several results for sparsely totient numbers. First, we show that a squarefree integer divides all sufficiently large sparsely totient numbers and a non-squarefree integer divides infinitely many sparsely totient numbers. Next, we construct explicit infinite families of sparsely totient numbers and describe their relationship with the distribution of consecutive primes. We also study the sparseness of $N_1$ and prove that it is multiplicatively piecewise syndetic but not additively piecewise syndetic. Finally, we investigate arithmetic/geometric progressions and other additive and multiplicative patterns like $\{x, y, x+y\}, \{x, y, xy\}, \{x+y, xy\}$ and their generalizations in the sparsely totient numbers.