TITLE: Highly composite number
QUESTION [7 upvotes]: Definition: n is said to be a highly composite number if and only if $d(n)>d(m)$ for all $m<n$, where $d(n)$ denotes number of divisors of n.
Questions:
1) Are there any theorems about highest prime that divides n, assuming that n is highly composite?. It seems that the highest prime is of the order of ln(n).
2) Are there any theorems about the highest power of 2 which divides n, assuming that n is highly composite?

REPLY [5 votes]: In this paper On Highly Composite Numbers P. Erdős uses Bertrand's postulate and the Prime Number Theorem to show that if $p$ is the highest prime that divides $n,$ a highly composite number, then
$$ c_1 \log n < p < c_2 \log n $$
for some constants $c_1$ and $c_2 .$