The literature refers to *smooth* integers as \begin{equation}\Psi(x,y):=\#\{n\le x:P_1(n)\le y\},\end{equation} where $P_1(n)$ is the largest prime factor of $n$. There are lots of results studying $\Psi(x,y)$, but I am interested in a least prime factors analogue. Is there a function, similar to $\Psi$, which counts $\{n\le x:p_1(n)\ge y\}$ or $\{n\le x:p_1(n)<y\}$?

I want to find a good estimate for $\#\{n\le x:p_1(n)\le m\}$; such an estimate exists for $P_1$ but I cannot find a similar one for $p_1$.