<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:
        n = len(nums)
        if n==1:
            return nums[0]
        squarefree = {1:([1]),2:([2])}
        primes = ([2])
        for i in range(3,n+1):
            prime = True
            sfree = True
            factor = 1
            t = i
            if self.isSquare(i):
                squarefree[1].append(i)
            else:
                for k in primes:
                    if t<k:
                        break
                    if (t%k)==0:
                        prime = False
                        power = 1
                        t = t//k
                        while t%k==0:
                            t = t//k
                            power+=1
                            sfree = False
                        if power%2==1:
                            factor*= k
                if not(sfree):
                    squarefree[factor].append(i)
                else:
                    squarefree[i] = [i]
                if (prime):
                    primes.append(i)
        sums = 0
        for key in squarefree.keys():
            sums = max(sums, sum([nums[t-1] for t in squarefree[key]]))
        return sums
        
    def isSquare(self, k):
        return (k==math.floor(math.sqrt(k))**2)
            
        
18800