By a theorem of Landau, the number of integers $n\leq x$ whose prime divisors belong to only arithmetic progressions $a_1,\dots,a_r$ mod $q$, with $r\leq\varphi(q)$ and $a_i$ coprime to $q$ for each $i$, is

$$Cx (\log x)^{r/\varphi(q)-1} + O(x (\log x)^{r/\varphi(q)-2}),$$

for some constant $C$ depending on $r$ and $q$.

This is an old result. I am sure much more is known about the distribution of such $n$, but I do not know the literature well enough, and would be happy if someone could provide suggestions regarding the state of the art (or other classical theorems) regarding results of this nature. Thanks!