Consider the function $\sigma(n)/n$, where $\sigma$ is the usual sum-of-divisors function. I read somewhere that it is unknown what rational numbers are in fact values of this function (or at any rate that characterizing them is an open question). Well, that was a while ago, and I suspect it was in one of my older references. So:

What is the current status of this question - characterizing the $q\in \mathbb{Q}$ such that there exists $n\in \mathbb{N}$ with $\sigma(n)/n=q$?

I think there is a standard name for the function $\sigma(n)/n$. If I knew it, that would make things easier, so I apologize if this is easy to find once one knows that.

Edit after accepting answer: Of course, $\sigma(n)/n=\sigma_{-1}(n)$, but I don't know whether there is so much more information under that designation!