I am trying to understand the patterns in the number of divisors $d(n)$ of the highly composite numbers.
The numbers marked with an asterisk are the superior highly composite numbers.
The first observation I made is that, for $n$ being a highly composite number, $d(n)$ is very often equal to a highly composite number. If it isn't, then it's at least equal to a number which has a lot of divisors.
When $n$ is a superior highly composite number, than $d(n)$ is extremely likely to be equal to a highly composite number: the first exception is $n=120$, and the second exception is $n=1441440$.
When $d(n)$ is equal to a highly composite number, than $n$ is extremely likely to be a superior highly composite number: the first exception is $d(n)=36$, and the second exception is $d(n)=180$.
The sequence of the values that $d(n)$ take, for $n$ being a highly composite number, includes every single highly composite number up to 720.
Something else also surprising is that $d(n)$ is sometimes equal to a power of $2$:
- d(2) = 2
- d(6) = 4
- d(24) = 8
- d(120) = 16
- d(840) = 32
- d(7560) = 64
- d(83160) = 128
- d(1081080) = 256
- d(17297280) = 512
- d(294053760) = 1024
I first thought that these numbers where simply the factorials (2, 6, 24, 120, ...), but unfortunately the $5^{th}$ term, 840, and every other following terms, break that pattern.
It is obvious that all of this is not a coincidence. There seems to be some kind of patterns governing all this... And my question is: what are those patterns?