If $d(n)$ denotes the number of divisors of $n \in \mathbb{N}$, we may define the function $$C(n) = \frac{\log(d(n)) \cdot \log(\log n)}{\log 2 \log n}.$$ 


According to Wikipedia, the Swedish mathematician Carl Severin Wigert proved that $\displaystyle \limsup C(n) = 1.$ $C$ can therefore reasonably be used as a measure of degree of compositeness. However, the natural log base is somewhat arbitrary, and arguably a better choice would be to use logs base 2, $\log\_2 n$, and set 
$$\displaystyle C\_b(n) = \frac{\log\_2(d(n)) \cdot \log\_2(\log\_2 n)}{\log\_2 n}.$$


This also has $\displaystyle \limsup$ equal to 1, and has the further nice property that $C\_b(2) = 0$, so that the minimal degree of compositeness is zero.


Suppose we define an "ultracomposite number" to be an $n \in \mathbb{N}$ such that $C\_b(n) > 1$, and for every $m < n$, we have $C\_b(m) < C\_b(n).$ 


It follows that the set of ultracomposite numbers is finite. Two questions now arise:


(1) What is the largest ultracomposite number, or failing that, what is a bound or estimate?


(2) How large is the set of ultracomposite numbers, or at least what is a bound or estimate for the size?


(3) More specifically, I conjecture there are seventeen ultracomposite numbers, of which 55440 is the largest.


It should be noted that the interest of this question depends on the acceptance of log base 2 as the privileged choice for this problem--natural logs, for example, lead to far more ultracomposites.



"Ultracomposite" numbers
EDIT, 11:36 pm. The conjecture is correct. From the first result of Nicolas and Robin,
$$ C\_2(n) \leq 1.53793986 + \frac{0.56367483}{\log \log n}. $$
This bound decreases with $n.$ Furthermore this gives
$ 1.743557976 $ when $n = 5433960.$ This says that we need only confirm the value of the ratio $C\_2(n)$ for $n \leq 5433960,$ by any means that brings joy, because $$ C\_2(55440) = 1.743557976.$$ 


ORIGINAL: Did it myself. For logs base 2 as described, 55440 now seems a good bet. For natural logarithms, Nicolas and Robin's number 6,983,776,800. Note where the "ratio" 1.537939861 appears in the first Nicolas-Robin result in my earlier answer. That is, it is a theorem that the relevant ratio attains its maximum at 6,983,776,800 and no other place. It may well be that the apparent list for the logs base 2 come as a corollary of one of the Nicolas-Robin results. 



```
  logs base 2:

  n   2 divisors 2 ratio   0   2 bump   2^1
  n   6 divisors 4 ratio   1.060087604   6 bump   3^1
  n   12 divisors 6 ratio   1.32815683   12 bump   2^2
  n   60 divisors 12 ratio   1.555150513   60 bump   5^1
  n   120 divisors 16 ratio   1.614640529   120 bump   2^3
  n   360 divisors 24 ratio   1.666250961   360 bump   3^2
  n   2520 divisors 48 ratio   1.729062351   2520 bump   7^1
  n   5040 divisors 60 ratio   1.73879969   5040 bump   2^4
  n   55440 divisors 120 ratio   1.743557976   55440 bump   11^1

natural logs:

  n   2 divisors 2 ratio   -0.5287663729   2 bump   2^1
  n   6 divisors 4 ratio   0.6509780925   6 bump   3^1
  n   12 divisors 6 ratio   0.9468861067   12 bump   2^2
  n   60 divisors 12 ratio   1.234235881   60 bump   5^1
  n   120 divisors 16 ratio   1.308415107   120 bump   2^3
  n   360 divisors 24 ratio   1.380756857   360 bump   3^2
  n   2520 divisors 48 ratio   1.467704178   2520 bump   7^1
  n   5040 divisors 60 ratio   1.484851215   5040 bump   2^4
  n   55440 divisors 120 ratio   1.51180374   55440 bump   11^1
  n   720720 divisors 240 ratio   1.525218379   720720 bump   13^1
  n   1441440 divisors 288 ratio   1.527797982   1441440 bump   2^5
  n   4324320 divisors 384 ratio   1.531905251   4324320 bump   3^3
  n   21621600 divisors 576 ratio   1.534731579   21621600 bump   5^2
  n   367567200 divisors 1152 ratio   1.537551622   367567200 bump   17^1
  n   6983776800 divisors 2304 ratio   1.537939861   6983776800 bump   19^1

```

