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Here are upper bounds 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, but this problem is somehow not quite equivalent, as shown by $24480$ for $n=8$ rather than the correct(?) minimum value of $3712$.

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.

Here are upper bounds 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, but this problem is not quite equivalent, as shown by $24480$ for $n=8$ rather than the correct minimum value of $3712$.

Here are upper bounds 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, but this problem is somehow not quite equivalent, as shown by $24480$ for $n=8$ rather than the correct(?) minimum value of $3712$.

Here are optimal solutions for $n \le 10$, and they happen to be equicardinal even if you don't enforce it: \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}

Here are upper bounds 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, but this problem is not quite equivalent, as shown by $24480$ for $n=8$ rather than the correct minimum value of $3712$.