# 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.

• Interesting question! It is probably not possible to give an explicit recipe. There are estimates for the number of times the most popular number appears, since there are estimates for the maximum value of $d(k)$ as $k$ ranges over the natural numbers $\le n$. – André Nicolas May 1 '14 at 1:31
• It seems somewhat rare for this question to have a unique answer. – David May 1 '14 at 1:50
• oeis.org/A057144 seems to be closer to the numbers asked for. – Gerry Myerson May 1 '14 at 12:51
• Not quite. For the solution x, you don't count the number of factors, you count the number of factors f with x/n <= f <= n. I suppose you can go through the numbers <= n with maximum number of factors and find an x with a large number of factors in the right range, then examine all x with at least that many factors, hoping there are not too many. – gnasher729 May 3 '14 at 19:26
• Based on a numerical experiment, it seems that these "most-often-occurring" numbers often are highly composite numbers. It's not true in all cases, though: For $n=1120$ we get $30240$, which is not a HCN; although this exception is apparently the sole one with $n\leq 4.10^4$. Furthermore, even this exceptional case happens to be product of primorials (just like all HCNs are) -- perhaps that could be a property shared by all the minimal-most-often-occurring numbers? – Peter Košinár May 11 '14 at 23:28

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.