I'm certain that you are aware of the fact that a number n is a prime only if
$$p(n) =\frac{\Gamma(n){^n}}{n} \notin \Bbb N$$
Now consider the function
$$\phi(x) = \begin{cases}1, & \text{if $x$ $\notin \Bbb N$}\\ 0, &\text{if $x \in \Bbb N$} \end{cases} = \lceil x - \lfloor x\rfloor \rceil $$
Using the above two functions, we have,
$$Q(x) = \begin{cases}1, & \text{if $x$ is a prime}\\ 0, &\text{otherwise } \end{cases} = \phi(p(x))$$
then the prime counting function $\pi(n)$ will be,
$$\pi(n) = \sum_{i= 2}^n \phi(p(i)) = \left\lceil \frac{\Gamma(i){^i}}{i} - \left\lfloor \frac{\Gamma(i){^i}}{i}\right\rfloor \right\rceil$$
Am I correct? Is there any way to proceed furthur?