<p>You are given a <strong>1</strong><strong>-indexed</strong> array <code>nums</code>. Your task is to select a <strong>complete subset</strong> from <code>nums</code> where every pair of selected indices multiplied is a <span data-keyword="perfect-square">perfect square,</span>. i. e. if you select <code>a<sub>i</sub></code> and <code>a<sub>j</sub></code>, <code>i * j</code> must be a perfect square.</p>

<p>Return the <em>sum</em> of the complete subset with the <em>maximum sum</em>.</p>

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

<div class="example-block">
<p><strong>Input:</strong> <span class="example-io">nums = [8,7,3,5,7,2,4,9]</span></p>

<p><strong>Output:</strong> <span class="example-io">16</span></p>

<p><strong>Explanation:</strong></p>

<p>We select elements at indices 2 and 8 and 2<code>&nbsp;* 8</code>&nbsp;is a perfect square.</p>
</div>

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

<div class="example-block">
<p><strong>Input:</strong> <span class="example-io">nums = [8,10,3,8,1,13,7,9,4]</span></p>

<p><strong>Output:</strong> <span class="example-io">20</span></p>

<p><strong>Explanation:</strong></p>

<p>We select elements at indices 1, 4, and 9. <code>1 * 4</code>, <code>1 * 9</code>, <code>4 * 9</code> are perfect squares.</p>
</div>

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

<ul>
	<li><code>1 &lt;= n == nums.length &lt;= 10<sup>4</sup></code></li>
	<li><code>1 &lt;= nums[i] &lt;= 10<sup>9</sup></code></li>
</ul>

class Solution:
    def maximumSum(self, nums: List[int]) -> int:
        # complete subset
        # every pair of selected values is a square?
        def findProdOddPrimes(num):
            prod = 1
            if num % 2 == 0:
                factor = 0
                while num % 2 == 0:
                    factor += 1
                    num //= 2
                prod = 2 if factor % 2 == 1 else 1
            lim = floor(sqrt(num))
            for i in range(3, lim+1, 2):
                if num % i == 0:
                    factor = 0
                    while num % i == 0:
                        factor += 1
                        num //= i
                    prod = prod * i if factor % 2 == 1 else prod
            # num is either 1 or a prime #
            return prod * num

        prodprimes = [0] * len(nums)
        for i in range(len(nums)):
            prodprimes[i] = findProdOddPrimes(i+1)
        
        # partition them into subsets where all values are equal
        sets = defaultdict(list)
        for i in range(len(nums)):
            sets[prodprimes[i]].append(nums[i])
        
        maxSum = 0
        for k,v in sets.items():
            # print(k, v)
            # if len(v) < 2 and k != 1:
            #     # only 1 element, cannot be a perfect square
            #     continue

            maxSum = max(maxSum, sum(v))
        
        # print(prodprimes)
        # print(sets)
        return maxSum
435