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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?

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  • $\begingroup$ Isn't $Z$ just equal to $X$? By definition, the elements of $Y$ are prime numbers, and so $m$ not being prime forces $m \not\in Y$. $\endgroup$ – Jeremy Rouse Jul 17 '15 at 12:33
  • $\begingroup$ @JeremyRouse: I read the condition as “$Y$ be a subset of $X$ containing all prime numbers, and possibly some other numbers”. $\endgroup$ – Emil Jeřábek Jul 17 '15 at 13:08
  • $\begingroup$ Yes, I have very little information on $Y$. I know that $Y$ is contained in $X$ and that it contains all prime numbers. Moreover I know that $Y$ is quite "small" compared to $X$. So, I am hoping that this will force $Z$ to be as "big" as $X$. (Answering to the clarification above: $Y$ can contain non prime numbers.) $\endgroup$ – Pablo Spiga Jul 17 '15 at 16:23
  • $\begingroup$ It might clarify things to note that this is "happening inside S", where S is the set of squarefree numbers and has a natural semilattice structure on it. Z is just X intersect the complement of a join-ideal of S generated by those members of Y that are not primes. If there are finitely many nonprime members of Y, it may be straightforward to show the ideal grows slowly. It may also be possible to find a "sparse infinite cover" which would force Z to be much smaller than you hope. Gerhard "This Mindset Worked Once Before" Paseman, 2015.07.17 $\endgroup$ – Gerhard Paseman Jul 17 '15 at 19:08

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