Infinitely many positive integers $n$ such that $\phi(n) = \frac{n}{4}$?

Question

Do there exist infinitely many positive integers $n$ such that $\phi(n) = \dfrac{n}{4}$?

Answer 1

There cannot be a single, let alone infinite, such $n$. Suppose that we write $4k = n$. Then we have $$\phi(4k) = k$$ Let $2^m$ be the highest power of $2$ dividing $k$. Then we can write $k = 2^mk'$ where $k'$ is odd. From this we get $$\phi(4k) = \phi(2^{m+2})\phi(k') = 2^mk'$$ which simplifies into $$2\phi(k') = k'$$ contradicting the fact that $k'$ is odd.

Answer 2

We have the well known formula that $$ \frac{\phi(n)}{n} = \prod_{p\mid n} \left(1-\frac{1}{p}\right);$$

from here it is easy to see the equation $$ \phi(n)=\frac{n}{k}$$

is only solvable for $k\le3$.