# Estimating the number of composite numbers less than a given $x$ that are relatively prime $6$

In a previous post, it was explained that the number of integers relatively prime to $6$ that are below a given $x$ is roughly $\frac{x}{3}$.

I am wondering if it is possible use this result to estimate the number of composite integers relatively prime to $6$ that are below a given $x$.

The goal would be to count the number of ways that these $\frac{x}{3}$ integers can be paired up.

To do this estimation, we would only need to consider how many ways the first $\frac{\sqrt{x}}{3}-1$ integers relatively prime to $6$ (not including $1$) pair up.

For example if $x = 144$, we would only need to consider the first $\frac{\sqrt{144}}{3} - 1 = 3$ which are $\{ 5, 7, 11\} < \sqrt{144}$

$5$ pairs up with $\frac{144}{5}\left(\frac{1}{3}\right)-1 = 8$ which are $\{5,7,11,13,17,19,23,25\}$ with each $< \frac{144}{5}$

$7$ pairs up with $\frac{144}{7}\left(\frac{1}{3}\right)-1 = 5$ which are $\{7, 11, 13, 17, 19\}$ with each $< \frac{144}{7}$

$11$ pairs up with $\frac{144}{11}\left(\frac{1}{3}\right)-1 = 2$ which are $\{11, 13\}$ with each $< \frac{144}{11}$

Does this approach make any sense? Is there a way to estimate the number of pairings for a given $x$?

## 1 Answer

All of the primes less than $x$, except exactly two (namely $2$ and $3$) are coprime to $6$. So to get the number of composite numbers less than $x$ that are coprime to $6$ all you need to do is estimate the number of coprimes (close to $x/3$) and subtract the number of primes less than $x$, minus 2, and then subtract one more because $1$ is usually considered neither prime not composite.

The number of primes less than $x$ can be approximated using the prime number theorem and its various variants and strenghtenings.