Let $n\in\mathbb{N}\setminus\{0\}$. Consider the summation over primes $p$ less than or equal to $n$
$$\psi(n)=\sum_{\substack{p\le n \\ p\; \text{prime}}}\left \lfloor{\frac{n}{p}}\right \rfloor. $$
Is $\psi(n)$ a well known function? If so what properties does it have, and in particular does it have a closed form?