Splitting the integers from $1$ to $2n$ into two sets with products as close as possible For each positive integer $n$, split the integers $1$ to $2n$ into two sets of $n$ elements each, and such that the products of the elements in each of these sets are as close as possible, say they differ by $a(n)$.
It can be checked that $a(1)=1$, $a(2)=2$, $a(3)=6$, $a(4)=18$, and $a(5)=30$.
Is this sequence strictly increasing?
What about if it is not required that the two sets contain the same number of elements?

Here are optimal solutions for $n \le 10$, and the two sets happen to be equicardinal even if you don't enforce that:
\begin{matrix}
n & a_n & \text{solution} \\
\hline
1 & 1 & \{2\},\{1\} \\
2 & 2 & \{2,3\},\{1,4\} \\
3 & 6 & \{1,5,6\},\{2,3,4\} \\
4 & 18 & \{1,5,6,7\},\{2,3,4,8\} \\
5 & 30 & \{2,3,4,8,10\},\{1,5,6,7,9\} \\
6 & 576 & \{1,4,7,8,9,11\},\{2,3,5,6,10,12\} \\
7 & 840 & \{2,4,5,6,8,11,14\},\{1,3,7,9,10,12,13\} \\
8 & 24480 & \{1,5,6,7,8,13,14,15\},\{2,3,4,9,10,11,12,16\} \\
9 & 93696 & \{2,3,6,8,9,11,12,13,18\},\{1,4,5,7,10,14,15,16,17\} \\
10 & 800640 & \{2,3,4,8,9,11,12,18,19,20\},\{1,5,6,7,10,13,14,15,16,17\} \\
\end{matrix}
I obtained these via integer linear programming by instead minimizing the difference in log products (sum of logs) rather than difference in products.
