For every positive integer $m$ let $f(m)=n$ be the $m$-th highly composite number. A positive integer is called highly-composite if it has more divisors then every smaller positive integer.
Suppose , some positive integer $k$ is given.
Can we efficiently determine
- whether there is some positive integer $l$ with $f(l)=k$ ? In other words , can we efficiently find out whether $k$ is a highly composite number ?
- If $k$ is highly composite , can we determine efficiently the positive integer $l$ ? In other words , can we find out the number in the numbered list of highly composite numbers ?
I am aware of necessary conditions for $k$ to be highly composite , but not of sufficient conditions. For the determination of $l$ , I have no idea at all. Has all this been worked out by someone ?
In OEIS , the first $10\ 000$ highly composite numbers are listed , but I am looking for a general method also working for much larger numbers.