Some questions about highly composite numbers

Question

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 ?

Finally

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 ?

Answer 1

Answer to the second question:

Suppose that $n>1680$ is highly composite and is not a multiple of $9$. Then: $$n=2^r\cdot 3\cdot 5\cdot\ldots\cdot p_k$$ where $p_k\ge11$, that is, $k\ge 5$

Then $t(n)=(r+1)2^{k-1}$. Define

$$n'=\frac{9n}{p_k}<n$$

Then $$t(n')=(r+1)4\cdot2^{k-3}=t(n)$$

which contradicts the fact that $n$ is highly composite.

Answer 2

You do not seem to be aware of the Superior Highly Composite Numbers. This is a subsequence defined by Ramanujan. One may find arbitrarily many of these, in order, by careful programming. I have done this. So, the largest known HC number is some extraordinarily large SHC number.

Meanwhile, G. Robin designed a method (1983) for using two consecutive SHC numbers and filling in all the HC numbers in between. So, fairly difficult task.

Answer 3

Answer to question 1:

Suppose we have $n = 2^{a_1} 3^{a_2} \ldots p_k^{a_k}$ with $a_1 \ge a_2 \ge \ldots \ge a_k \ge 2$, $n > 36$ and $n$ is highly composite. I claim that $a_1 \ge 3$. Suppose $a_1 \le 2$; if $k=2$, then $n \le 2^2 3^2 = 36$, whereas if $k \ge 3$, then $t(\frac{4n}{5}) = \frac{a_1+3}{a_1+1} \cdot \frac{a_3}{a_3+1}\cdot t(n) \ge \frac{5}{3}\cdot\frac{2}{3}\cdot t(n) > t(n)$, so $n$ is not highly composite.

But then $t(\frac{p_{k+1} n}{2 p_k}) = 2 \cdot \frac{a_1}{a_1+1} \cdot \frac{a_k}{a_k+1}\cdot t(n) \ge 2 \cdot \frac{3}{4} \cdot \frac{2}{3} \cdot t(n) = t(n)$, while $\frac{p_{k+1} n}{2 p_k} < n$ by Bertrand's Postulate.