<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:
        '''
        100,000
        
        the indice products are perfect squares
        
        1, 4 = 4
        
        2, 8 = 16
        
        1, 9 = 16
        
        1, 4, 9 = 14
        
        goal: get length of nums N
        consider the array 1-indexed
        
        get every comination of numbers 1...n s.t. the product is a perfect square in nlogntime
        
        sum these possibilities and take largest nlogntime
        
        all numbers in the array are positive, so the "more" numbers the better
        
        ex: if the array is length 9 or longer, we dont care about 1, 4, 9, or 1 4, or 1, 9 only 1, 4, 9
        
        1, 4, 9, 16

        2, 8, 18, 32

        3, 12, 27, 48
        ...
        '''

        ans = max(nums)
        N = len(nums)

        
        for start in range(1, N):
            total = 0
            a = 1

            while a * a * start <= N:
                total += nums[a * a * start- 1]
                a += 1


            ans = max(ans, total)
    
    

        
        return ans
246