<p>You are given an integer array of unique positive integers <code>nums</code>. Consider the following graph:</p>

<ul>
	<li>There are <code>nums.length</code> nodes, labeled <code>nums[0]</code> to <code>nums[nums.length - 1]</code>,</li>
	<li>There is an undirected edge between <code>nums[i]</code> and <code>nums[j]</code> if <code>nums[i]</code> and <code>nums[j]</code> share a common factor greater than <code>1</code>.</li>
</ul>

<p>Return <em>the size of the largest connected component in the graph</em>.</p>

<p>&nbsp;</p>
<p><strong class="example">Example 1:</strong></p>
<img alt="" src="https://assets.leetcode.com/uploads/2018/12/01/ex1.png" style="width: 500px; height: 97px;" />
<pre>
<strong>Input:</strong> nums = [4,6,15,35]
<strong>Output:</strong> 4
</pre>

<p><strong class="example">Example 2:</strong></p>
<img alt="" src="https://assets.leetcode.com/uploads/2018/12/01/ex2.png" style="width: 500px; height: 85px;" />
<pre>
<strong>Input:</strong> nums = [20,50,9,63]
<strong>Output:</strong> 2
</pre>

<p><strong class="example">Example 3:</strong></p>
<img alt="" src="https://assets.leetcode.com/uploads/2018/12/01/ex3.png" style="width: 500px; height: 260px;" />
<pre>
<strong>Input:</strong> nums = [2,3,6,7,4,12,21,39]
<strong>Output:</strong> 8
</pre>

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

<ul>
	<li><code>1 &lt;= nums.length &lt;= 2 * 10<sup>4</sup></code></li>
	<li><code>1 &lt;= nums[i] &lt;= 10<sup>5</sup></code></li>
	<li>All the values of <code>nums</code> are <strong>unique</strong>.</li>
</ul>

class Solution:
    def largestComponentSize(self, nums: List[int]) -> int:
        def factors(n):
            if n % 2 == 0:
                yield 2
            while n % 2 == 0:
                n //= 2
            for i in range(3, isqrt(n) + 1, 2):
                while n % i == 0:
                    yield i
                    n //= i
            if n > 2:
                yield n

        sets = {i: {i} for i in range(len(nums))}
        not_parent = set()

        parents = [i for i in range(len(nums))]
        def find(i):
            while i != parents[i]:
                i = parents[i] = parents[parents[i]]
            return i

        rank = [1 for _ in range(len(nums))]
        def union(i, j):
            pi, pj = find(i), find(j)
            if pi == pj:
                return

            expected_size = len(sets[pi]) + len(sets[pj])
            new_set = sets[pi] | sets[pj]
            if rank[pi] > rank[pj]:
                not_parent.add(pj)
                sets[pi] |= sets[pj]
                parents[pj] = pi
                rank[pi] += rank[pj]
                rank[pj] = 0
            else:
                not_parent.add(pi)
                sets[pj] |= sets[pi]
                parents[pi] = pj
                rank[pj] += rank[pi]
                rank[pi] = 0


        def print_sets():
            sets = defaultdict(set)
            for i, n in enumerate(nums):
                sets[find(i)].add(n)
            for p, s in sets.items():
                print((p, rank[p], len(s), s))

        prime_map = {}
        for i, n in enumerate(nums):
            for prime in factors(n):
                if prime in prime_map:
                    union(prime_map[prime], i)
                else:
                    prime_map[prime] = i
        
        #print_sets()

        #print(len(nums))
        #print(sum(rank[i] for i in range(len(nums)) if i == find(i)))
        #print(max(rank))

        return max(rank)

31065