Let $\mathbb{N}$ denote the set of positive integers. For $a\in \mathbb{N}$ let $p(a)$ be the number of prime divisors of $a$. So we have $p(1)=0$, and $p(n) > 1$ for $n\in\mathbb{N}\setminus\{1\}$.

For $n\in \mathbb{N}$, let $\text{med}(n)$ be the median of the set $\{p(a): 1\leq a \leq n\}$.

Questions. Is the sequence $(\text{med}(n))_{n\in\mathbb{N}}$ bounded? If yes, does it have a limit, and is the (integer) value of the limit known?