Let $2\leq d\_1 < d\_2,...,d\_l < n$ be all the proper nontrivial divisors of $n$. I like to understand how much these divisors deviates from each other. Here are two questions in this regard:


(1) What is the maximum of the set $\{d\_i/d\_{i-1}: 1\leq i \leq l\}$. Say it $M$.


Assume that you know the prime factorization of $n (= p\_1^{\alpha\_1}..p\_r^{\alpha\_r})$. Can I have a formula in terms of $p\_i$'s and $\alpha\_i$'s. One of the crude upper bound can be $p\_r$ but this is really bad if $n$ has many distinct prime factors. 


(2) Instead of maximum, the average may be more interesting and useful. So can we estimate 
the mean and variance,


$\mu = (1/l)\sum\_{i=1}^{l}{d\_i/d\_{i-1}}$ and $\sigma^2 = (1/l)\sum\_{i=1}^{l}{(d\_i/d\_{i-1}-\mu)^2}$



Ratio of consecutive divisors and average 
Tenenbaum [Sur un probleme de crible et ses applications (1986)] showed that 
$$ F(n)/n=\max\_{1\le i < \tau(n)} \frac{d\_{i+1}(n)}{d\_i(n)},$$
where $1=d\_1(n)< \ldots < d\_{\tau(n)}(n)=n$ is the increasing sequence of divisors, and 
$F$ is given by $F(1)=1$ and $F(n)=\max \{ d P^{-}(d) : d|n,\, d>1 \}$ for $n\ge 2$, where $P^{-}(n)$ is the smallest prime divisor of $n$. 
For $n=p\_1^{\alpha\_1}\cdots p\_k^{\alpha\_k}$, with $p\_1<\ldots < p\_k$, this leads to the formula
$$ \max\_{1\le i < \tau(n)} \frac{d\_{i+1}(n)}{d\_i(n)}
= \max\_{1 \le j \le k} \{ p\_j p\_j^{\alpha\_j}\cdots p\_k^{\alpha\_k} \}/n$$


Let $D(x,t)=|\{n\le x: F(n)/n \le t\}|$. Saias [Entiers a diviseurs denses 1 (1997)] showed that $D(x,t) \asymp x\log t /\log x$ for $x\ge t \ge 2$. I showed [Integers with dense divisors, 2 (2004)] that $D(x,t)=x\, d(w) \{1+O(1/\log t) \}$, for $x\ge 3$, $x\ge t \ge \exp\{(\log(\log(x)))^{5/3+\varepsilon}\}$, where $w=\log x / \log t$ and $d(w)$ is a continuous function that can be expressed in terms of Dickman's function and which satisfies $d(w) \asymp 1/w$. 


2015 Update: In [arXiv:1405.2585], I found that $D(x,t)=x\, d(w) \{1+O(1/\log t) \}$, for $x\ge t \ge 2$. 


