A highly composite number is a natural number $n\ge 1$, such that $t(m)<t(n)$ for all $m$ with $1\le m<n$ , where $t(n)$ is the number of divisors of $n$.
The link shows that the prime factorization of such a number $n$ contains non-increasing exponents and contains all primes upto the largest prime factor of $n$.
It is also claimed that for every $n>36$ the last exponent is always $1$, in other words, if $p$ is the largest prime factor of a highly composite number $n>36$, then $p^2$ does not divide $n$.
How can I prove this ?
The largest highly composite number not divisible by $9$, seems to be $$1680=2^4\cdot 3\cdot 5\cdot 7$$
Is this true, and how can I prove it ?
Given the sequence of the exponents, is there an efficient method to determine whether the sequence corresponds to a highly composite number ?
It is clear that it is sufficient to test the numbers corresponding to a non-increasing sequence and that the length is bounded by the smallest primorial exceeding the given number $n$. But for large $n$, many such sequences have to be checked.
Just for the sake of curiousity : What is the largest known highly composite number ?