<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
import math

class Solution:
    def largestComponentSize(self, nums: List[int]) -> int:
        max_num = max(nums)
        
        spf = list(range(max_num + 1))
        for i in range(2, int(math.sqrt(max_num)) + 1):
            if spf[i] == i:
                for j in range(i * i, max_num + 1, i):
                    if spf[j] == j:
                        spf[j] = i
        
        def get_prime_factors(x):
            factors = set()
            while x > 1:
                factors.add(spf[x])
                x //= spf[x]
            return factors

        parent = list(range(max_num + 1))
        rank = [1] * (max_num + 1)
        
        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 rank[rootX] > rank[rootY]:
                    parent[rootY] = rootX
                elif rank[rootX] < rank[rootY]:
                    parent[rootX] = rootY
                else:
                    parent[rootY] = rootX
                    rank[rootX] += 1
        
        prime_to_index = defaultdict(list)
        
        for num in nums:
            prime_factors = get_prime_factors(num)
            for prime in prime_factors:
                prime_to_index[prime].append(num)
        
        for prime, numbers in prime_to_index.items():
            first_num = numbers[0]
            for num in numbers[1:]:
                union(first_num, num)
        
        component_size = defaultdict(int)
        for num in nums:
            root = find(num)
            component_size[root] += 1
        
        return max(component_size.values())

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