# Cardinality of a floor function set

## TL;DR

This paper investigates the size of the set formed by the floor division of a positive integer X by all integers from 1 to X, including extensions to subsets like primes and semiprimes.

## Contribution

It provides new bounds and insights into the cardinality of floor function sets and their prime-related subsets for a fixed positive integer.

## Key findings

- Derived bounds for the set size
- Analysis of prime and semiprime subsets
- Extensions to related number sets

## Abstract

Fix a positive integer X. We quantify the cardinality of the set $\{\lfloor X/n \rfloor\}_{n=1}^X$. We discuss restricting the set to those elements that are prime, semiprime or similar.