What number appears most often in an $n \times n$ multiplication table?

The question is precisely as stated in the title:

What number appears most often in an $n \times n$ multiplication table?

Note: By "an $n \times n$ multiplication table" I mean the multiset

$$M_n := \{a \cdot b: \mathbb{Z}^{+}\ni a, b \leq n \} $$

I realize the answer is often not unique - though one could make it so by asking for the minimal entry in the case of a tie - but I am wondering whether there is a general approach to this question.

I am not sure about how difficult this problem is; for example, a related question about distinct entries turns out to be quite nontrivial: See the discussion of the Erdos Multiplication Table problem, which was formulated in the mid-twentieth century and resolved only recently by Ford (2008), in the MathOverflow post here.

Solution 1:

Here are some experimental data. I just used brute force to compute the (smallest) most occuring number $a_n$ and its multiplicity $b_n$ for $1\leq n\leq 1000$. E.g., $$a_{1000}=27720=2^3\cdot 3^2\cdot5\cdot7\cdot11\ , \qquad b_{1000}=58\ .$$ The following figures show list plots of the $a_n$ and the $b_n$. Note that the $a_n$ are not monotone increasing.