TITLE: What is the lower bound for highly composite numbers?
QUESTION [10 upvotes]: if $x=d(n)$ is the number of divisors of $n$, what is the tightest lower-bound for $n$ only given $x$?
http://en.wikipedia.org/wiki/Highly_composite_number

REPLY [2 votes]: It would be nice to have an inequality $n \ge f(x)$. If the poser wants numerical results, here are two:
The least number having exactly x divisors is given by OEIS sequence http://www.oeis.org/A005179. It is a pretty wild function. The nice paper by Grost is recommended.
The least number having x (or more) divisors is given by the OEIS sequence http://www.oeis.org/A061799.