The rate of growth of small divisors of an integer \begin{align}
1 & \times 360 \\
2 & \times 180 \\
3 & \times 120 \\
4 & \times 90 \\
5 & \times 72 \\
6 & \times 60 \\
8 & \times 45 \\
9 & \times 40 \\
10 & \times 36 \\
12 & \times 30 \\
15 & \times 24 \\
18 & \times 20 \\
\uparrow
\end{align}
In this way of listing all divisors of a number, the ones $\le$ the square root are in the left column, and the biggest one has grown to size $18$ in $12$ steps, thus at an average rate of $18/12=1.5$.
Trying this with another number, we have
\begin{align}
1 & \times 360360 \\
2 & \times 180180 \\
3 & \times 120120 \\
4 & \times 90090 \\
5 & \times 72072 \\
6 & \times 60060 \\
7 & \times 51480 \\
8 & \times 45045 \\
9 & \times 40040 \\
10 & \times 36036 \\
11 & \times 32760 \\
12 & \times 30030 \\
13 & \times 27720 \\
& {}\; \vdots \\
585 & \times 616
\end{align}
The average rate is $585/96= 6.09375$, which is quite a lot smaller than average for such a big number, if I'm not mistaken.
Is this average rate function "known"?  Does it have interesting properties?  Are there theorems of interest about it?
(Idle curiosity inspired by this page, a proposed Wikipedia article that has not received good reviews.)
 A: Ramanujan observed that the number of divisors of $N$ "varies with extreme irregularity, tending to infinity or remaining small according to the form of $N$."
There is an article$^1$ by him entitled Highly Composite Numbers in which he deals with this and related questions. He says most authors had focused on the average order of the number of divisors $d(N)$ and that Dirichlet proved$^2$ 
$$\frac{d(1)+d(2)+d(3)+\cdots+d(N)}{N}=\log N + 2\gamma-1+O\left(\frac 1{\sqrt{N}}\right).$$
Instead Ramanujan focuses on the maximum order of these numbers and cites Wigert$^{3}$ to the effect that
$$d(N)< 2^{(\log N/\log\log n)(1 + \varepsilon)} $$
Ramanujan then shows that 
$$d(N) < 2^{\operatorname{Li}(\log n)}+ O\left(\log N e^{-\alpha \sqrt{\log\log N}}\right). $$
$^1$ Ramanujan, Highly Composite Numbers, Collected Papers, pp. 79-128.
$^2$ Citing Werke vol.2, p. 49.
$^3$ Arkiv fur Matematik, vol. 3, p 18. 
A: The candidates are the highly composite numbers, which are OEIS A002182.  The number of factors less than $\sqrt n$ is half (unless a square) of A002183.  The best average is $4$, with factors $1,2$ less than or equal to $\sqrt {4}$ and a growth rate of $2/2=1$ There are some off by $1$ errors in your definition of growth rate compared to what one might expect-the number of gaps is one less than the number of factors and the growth might be one less as the smallest factor is $1$ but I have followed your definitions.  The next minimum is $36$, with $5$ factors less than or equal to $6$, so a growth rate of $1.2$  The growth rate seems to climb after that, so I think the interesting thing would be what is the asymptotic value of the inf of the growth rate for numbers greater than $n$.
