# Rough integers with a divisor in a given interval

## TL;DR

This paper estimates the count of integers up to x with specific divisor properties and applies these results to analyze the distribution of integers and fractions in multiplication tables, focusing on prime factor restrictions.

## Contribution

It provides uniform asymptotic estimates for integers with a divisor in a given interval and no small prime factors, extending to applications in multiplication tables and fraction counts.

## Key findings

- Estimated the number of integers with a divisor in (y, 2y] and no small prime factors.
- Applied estimates to count distinct integers in multiplication tables free of small primes.
- Analyzed the number of distinct fractions formed from bounded integers.

## Abstract

We determine, up to multiplicative constants, the number of integers $n\le x$ that have no prime factor $\le w$ and a divisor in $(y,2y]$. Our estimate is uniform in $x,y,w$. We apply this to determine the order of the number of distinct integers in the $N\times N$ multiplication table which are free of prime factors $\le w$, and the number of distinct fractions of the form $\frac{a_1a_2}{b_1b_2}$ with $1\le a_1 \le b_1\le N$ and $1\le a_2\le b_2 \le N$.