<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>

import math
class Solution:
    def maxScore(self, nums: List[int]) -> int:
        n = len(nums)
        m = n>>1
        done = (1<<(n))-1
        memo = [[0] * (m+1) for _ in range(done+1)]
        gcds = [[0] *n for _ in range(n)]
        for i in range(n):
            for j in range(i, n):
                val = math.gcd(nums[i],nums[j])
                gcds[i][j] = val
                gcds[j][i] = val
        
        def dp(memo, state, op):
            if op > m:
                return 0
            if memo[state][op]>0:
                return memo[state][op]

            max_partial = 0
            for i in range(n-1):
                for j in range(i+1, n):
                    if state & (1<<i) == 0 and state & (1<<j) == 0:
                        
                        new_state = state | (1<<i) | (1 <<j)
                        max_rest = dp(memo, new_state, op+1)
                        max_partial = max(max_partial, gcds[i][j]*op + max_rest)

            memo[state][op] = max_partial
            return memo[state][op]
                        

        dp(memo, 0, 1)
        
        return memo[0][1]


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