For the problem I am working on I have realised that some of the proofs could be slightly simplified if a certain number theoretic question has a positive solution. Now, the problem is in finite group theory, and unless the answer is really "immediate" I don't think that I intend to use number theoretic techniques. Nevertheless I am intrigued by the question and I would love to know the answer. Here it is.

Consider the set $X$ of natural numbers $n$ relatively prime to their Euler totien function $\varphi(n)$. It is important (I believe) to observe that, given $n\in\mathbb{N}$, the size of $\{x\in X\mid x\leq n\}$ is asymptotic to $an/\log\log\log n$ (for some absolute constant $a$).

Let $Y$ be a subset of $X$ containing all prime numbers. Suppose that, given $n$, the size of $\{y\in Y\mid y\leq n\}$ is asymptotic to $bn/\log n$ (for some absolute constant $b$). This is the whole information I have for $Y$.

Consider $Z$ the subset of $X$ consisting of all the elements $x\in X$ such that, for every divisor $m$ of $x$ with $m$ not being a prime, we have $m\notin Y$.

I am interested in the growth of the size of $\{z\in Z\mid z\leq n\}$. Does it grow as $n/\log\log\log n$ or less?