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. 



Distribution function for divisors of an Integer
The distribution of divisors on 'average' is known (so to say giving the behavior of a 'typical'integer).


Set $F\_n(u) = d(n)^{-1} D\_n(n^u)$ where $d(n)$ is the total number of divisors of $n$, then 


$$x^{-1} \sum\_{n\le x} F\_n(u) = \frac{2}{\pi} \arcsin \sqrt{u} + O((\log x)^{-1/2}) $$ 
uniformly for $x\ge 2$ and $u \in [0,1]$. 
This is due to Deshouillers, Dress, Tennenbaum (1979) and can be found for example in Tennenbaum's Introduction to Analytic and Probabilistic Number Theory (see Thm 7 in II.6). 


Roughly this means, there are many small divisors. 


A lot of information on this type of question, and I assume also this result but could not check, is in Hall and Tennenbaum's book 'Divisors'. 


