For a fixed $n$, let $D_n(x) = \{ d|n : d \leq x \}$ . We assume here $p \leq x \leq n/p$, where $p$ is the smallest prime factor of $n$.

For example if $n = p^i$ for some prime $p$ then $D_n(x) \sim \log x/ \log p$.

What are the other 'nice' distribution function of divisors that are satisfied by an infinite family of integers ( like $\log x/ \log p$ for $\{p^i, i \in \mathbb{N}\}$ ).

Here 'nice', means it is easy to do calculus (integrate or differentiate) with such functions.