<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>

from collections import defaultdict
from math import isqrt
from typing import List

class Solution:
    def largestComponentSize(self, nums: List[int]) -> int:
        def find(x):
            if parent[x] != x:
                parent[x] = find(parent[x])
            return parent[x]
        
        def union(x, y):
            rootX = find(x)
            rootY = find(y)
            if rootX != rootY:
                if size[rootX] > size[rootY]:
                    rootX, rootY = rootY, rootX
                parent[rootX] = rootY
                size[rootY] += size[rootX]
        
        def prime_factors(n):
            factors = set()
            while n % 2 == 0:
                factors.add(2)
                n //= 2
            for i in range(3, isqrt(n) + 1, 2):
                while n % i == 0:
                    factors.add(i)
                    n //= i
            if n > 2:
                factors.add(n)
            return factors
        
        # Union-Find Initialization
        parent = {}
        size = {}
        for num in nums:
            parent[num] = num
            size[num] = 1
        
        # Map from factor to the first number having this factor
        factor_map = defaultdict(int)
        
        for num in nums:
            factors = prime_factors(num)
            for factor in factors:
                if factor in factor_map:
                    union(num, factor_map[factor])
                factor_map[factor] = num
        
        # Calculate the size of each component
        component_size = defaultdict(int)
        for num in nums:
            root = find(num)
            component_size[root] += 1
        
        # The largest component size
        return max(component_size.values())

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