Let $\Phi(n) = \sum_{k=1}^n \phi(k)$ be the totient summatory function. Here is an interesting conjecture I've made: The ratio $\Phi(n^2)/\Phi(n)$ is an integer only for $n=1,2,3,5$ and $6$. I made a program that shows that these are only solution for about $n<3000$. Is there some way to prove or disaprove it?
Eventually, what about the general case: what conditions should satisfy numbers $n$ and $m$ so that ratio of their totient summatory functions is integer? Is there some paper that deals with it?