What's kind of integer n such that $\varphi(n)>n$?

Question

I want to know which integer $n$ has a property that $\varphi(n)>\dfrac{n}{2}$, where $\varphi(n)$ is the Euler's totient function.

Thank you.

Answer 1

We have Euler's product formula:

$$\displaystyle\varphi(n)=n\prod_{p\mathop\vert n}\left(1-\dfrac1p\right)$$

Therefore, your question will be equivalent to finding a set of primes $P$ such that:

$$\displaystyle\prod_{p\mathop\in P}\left(1-\frac1p\right)>\frac12$$ And then forming integers by using those primes only.

For example, one set that satisfies the above formula is $\{3,5\}$.

Therefore, any number of the form $3^a5^b$ where $a,b\ge1$ and $a,b\in\mathbb N$ will satisfy your question.

For example, $3^25^1=45$, and $\varphi(45)=24>\dfrac{45}2=22.5$.