Obviously, we do not know an explicit and easily manipulable formula for finding every prime - that is, a function $f(n)$ which yields the $n^{th}$ prime. I've seen plenty of formulas that "cheat" in some way (e.g. encode trial division in a sum and floor functions) or only yield a very sparse set of primes (e.g. Mills' formula).
I'm curious about results of the existence or non-existence of $f$ satisfying certain "sanity" conditions. What is clear to me is that:
There is no suitable rational function, because they are all in $O(x^k)$ for integer $k$, which is the wrong asymptotic.
There is an analytic function giving primes, because any sequence can be given by an analytic function.
However, these are really trivial observations, and they leave a very wide gap of functions for which we don't know much. Could there be any $f$ writable in terms of integrals of rational functions? Could there be an $f$ with some power series which converges quickly? Could there be some $f$ defined by an (algebraic) differential equation? I'd be interested in any (non-trivial*) result of existence or non-existence of such functions - or even about functions which always yields primes, but don't yield every prime.
(*i.e. "trivial" being a result that would hold for any sequence in $O(n\log(n))$)