Is there a known function $f(n) = P_n$, where $P_n$ denotes the $n$th prime number? 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

*$\forall n\in\mathbb{N}:f(n) = P_n$, 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?
 A: Your question seems to reflect a misunderstanding of what a function is.
Here are some examples of functions:
\begin{align*}
f(x) &= \text{the number of primes less than or equal to } x \\
f(x) &= 1 \text{ if the third digit of } x \text{ is even, otherwise } 2 \\
f(x) &= 1 \text{ if } x \text{ is rational, otherwise } 2
\end{align*}
A function simply means you take in an input and send out an output.
A function does not have to have a simple "formula" like $x^2$ or $e^x$.  In particular, it is extremely easy to make a function such that $f(n) = P_n$.  Just define $f(x)$ to be the $x$th prime if $x$ is an integer, and then define it arbitrarily elsewhere.

  
*
  
*Needless to say, the definition of $f$ does not make an explicit use of $P_n$
  

This and all your other constraints don't really make mathematical sense.  I can define $f(x)$ equals the product of all positive integers less than or equal to $x$.  This results essentially in the factorial function, but I am not using the $!$ symbol.  Or, I can define
$f(x) = \int_0^{\infty} e^{-t} t^n \; dt$, and again I have defined factorials without really using factorials.
There are ways to make your idea of a "function" precise, but it's a lot harder than you think.  For example, you can ask if there is an Elementary function satisfying $f(n) = P_n$.  But it sounds like you want a broader scope than just exponentials, $n$th roots, and polynomials.
Some related references:
Can insight be derived from direct formulae for prime numbers?
Prime number formulas
What would be the immediate implications of a formula for prime numbers?
A: Sure. Define $f(n)$ for positive integer $n$ as follows:
$$
f(n)=n\chi _{{{\mathbb  {P}}}}(n),
$$
where $\mathbb{P}$ denotes the set of prime numbers $\chi_\mathbb{P}$ is the characteristic function of the primes, i.e., the function such that for positive integer $n$:
$$
{\displaystyle \chi_\mathbb{P}(n):={\begin{cases}1&{\text{if }}x\in \mathbb{P},\\0&{\text{if }}x\notin \mathbb{P}.\end{cases}}}
$$
