<p>You are given <code>nums</code>, an array of positive integers of size <code>2 * n</code>. You must perform <code>n</code> operations on this array.</p>

<p>In the <code>i<sup>th</sup></code> operation <strong>(1-indexed)</strong>, you will:</p>

<ul>
	<li>Choose two elements, <code>x</code> and <code>y</code>.</li>
	<li>Receive a score of <code>i * gcd(x, y)</code>.</li>
	<li>Remove <code>x</code> and <code>y</code> from <code>nums</code>.</li>
</ul>

<p>Return <em>the maximum score you can receive after performing </em><code>n</code><em> operations.</em></p>

<p>The function <code>gcd(x, y)</code> is the greatest common divisor of <code>x</code> and <code>y</code>.</p>

<p>&nbsp;</p>
<p><strong class="example">Example 1:</strong></p>

<pre>
<strong>Input:</strong> nums = [1,2]
<strong>Output:</strong> 1
<strong>Explanation:</strong>&nbsp;The optimal choice of operations is:
(1 * gcd(1, 2)) = 1
</pre>

<p><strong class="example">Example 2:</strong></p>

<pre>
<strong>Input:</strong> nums = [3,4,6,8]
<strong>Output:</strong> 11
<strong>Explanation:</strong>&nbsp;The optimal choice of operations is:
(1 * gcd(3, 6)) + (2 * gcd(4, 8)) = 3 + 8 = 11
</pre>

<p><strong class="example">Example 3:</strong></p>

<pre>
<strong>Input:</strong> nums = [1,2,3,4,5,6]
<strong>Output:</strong> 14
<strong>Explanation:</strong>&nbsp;The optimal choice of operations is:
(1 * gcd(1, 5)) + (2 * gcd(2, 4)) + (3 * gcd(3, 6)) = 1 + 4 + 9 = 14
</pre>

<p>&nbsp;</p>
<p><strong>Constraints:</strong></p>

<ul>
	<li><code>1 &lt;= n &lt;= 7</code></li>
	<li><code>nums.length == 2 * n</code></li>
	<li><code>1 &lt;= nums[i] &lt;= 10<sup>6</sup></code></li>
</ul>

from math import gcd
from typing import List

class Solution:
    def maxScore(self, nums: List[int]) -> int:
        n = len(nums) // 2
        self.max_score = 0
        self.nums = nums
        self.memo = {}
        
        return self.dfs(0, 0, n)

    def dfs(self, mask: int, pairs_picked: int, n: int) -> int:
        # If we have already calculated the result for this mask, return it.
        if (mask, pairs_picked) in self.memo:
            return self.memo[(mask, pairs_picked)]
        
        # If we have picked all elements, return 0.
        if pairs_picked == n:
            return 0
        
        max_score = 0
        for i in range(2 * n):
            if mask & (1 << i):
                continue
            for j in range(i + 1, 2 * n):
                if not mask & (1 << j):
                    new_mask = mask | (1 << i) | (1 << j)
                    score = (pairs_picked + 1) * gcd(self.nums[i], self.nums[j])
                    max_score = max(max_score, score + self.dfs(new_mask, pairs_picked + 1, n))
        
        # Memoize the result for this mask.
        self.memo[(mask, pairs_picked)] = max_score
        return max_score

# Example usage:
solution = Solution()
print(solution.maxScore([1, 2]))  # Output: 1
print(solution.maxScore([3, 4, 6, 8]))  # Output: 11
print(solution.maxScore([1, 2, 3, 4, 5, 6]))  # Output: 14
18000