<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>

import collections
import functools
import itertools
import math
from typing import List


class DisjointSet:
    def __init__(self, n):
        self._group = list(range(n))
        self._rank = [0] * n
        self._group_size = [1] * n

    def union(self, a, b):
        if (a := self._find(a)) != (b := self._find(b)):
            if self._rank[a] < self._rank[b]:
                a, b = b, a
            self._group[b] = a
            self._rank[a] += self._rank[a] == self._rank[b]
            self._group_size[a] += self._group_size[b]

    def group_size(self, a):
        return self._group_size[self._find(a)]

    def _find(self, a):
        if self._group[a] != a:
            self._group[a] = self._find(self._group[a])
        return self._group[a]


class Solution:
    def largestComponentSize(self, nums: List[int]) -> int:
        @functools.cache
        def factorization(a):
            for p in range(2, int(math.sqrt(a)) + 1):
                if a % p == 0:
                    return factorization(a // p) | set([p])
            return set([a])

        prime_to_nodes = collections.defaultdict(list)
        for i, a in enumerate(nums):
            for p in factorization(a):
                prime_to_nodes[p].append(i)

        n = len(nums)
        ds = DisjointSet(n)
        for indices in prime_to_nodes.values():
            for a, b in itertools.pairwise(indices):
                ds.union(a, b)
        return max(ds.group_size(i) for i in range(n))

44824