Let $d(n)$ be the number of divisors of $n$. Let $p(n)$ be the product of the divisors of $n$. Ramanujan called a number $n$ largely composite if $d(n) \ge d(m)$ for $m < n$. Let's call $n$ fully composite if $p(n) \ge p(m)$ for $m < n$. It is conjectured that largely composite numbers (LCN) are the same as fully composite numbers (FCN). It is easy to show that an LCN is an FCN. Is every FCN an LCN? In the OEIS, these are sequences http://oeis.org/classic/A067128 and http://oeis.org/classic/A034287. These two sequences are the same for the 105834 terms less than 10^150.
This question is interesting because it connects the number of divisors to the product of divisors.