Is there a known function $f:\mathbb{R}\to\mathbb{R}$, such that:

* The definition of $f$ does not contain the $!$ operator
* The definition of $f$ does not contain the $\sum$ operator
* The definition of $f$ does not contain the $\prod$ operator
* The definition of $f$ does not contain a [continued fraction](http://en.wikipedia.org/wiki/Continued_fraction)
* $\forall n\in\mathbb{N}:f(n) = P\_n$, where $P\_n$ denotes the $n$th prime number
* Needless to say, the definition of $f$ does not make an explicit use of $P\_n$

If no such function is known, then is there any known proof that no such function exists?