Complete List of Special Highly Composite Numbers

Theorem
There are exactly $6$ special highly composite numbers:
 * $1, 2, 6, 12, 60, 2520$

Proof
We have the following:


 * $1$ is a Special Highly Composite Number


 * $2$ is a Special Highly Composite Number


 * $6$ is a Special Highly Composite Number


 * $12$ is a Special Highly Composite Number


 * $60$ is a Special Highly Composite Number


 * $2520$ is a Special Highly Composite Number

By inspection of the sequence of highly composite numbers:
 * $1, 2, 4, 6, 12, 24, 36, 48, 60, 120, 180, 240, 360, 720, 840, 1260, 1680, 2520, \ldots$

it can be seen that there are no more special highly composite numbers less than $2520$.

Let $n > 18$.

From Ratio between Consecutive Highly Composite Numbers Greater than 2520 is Less than 2, the $n$th highly composite number does not divide the $n+1$th.

Hence the $n$th highly composite number is not a special highly composite number.

The result follows.