<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 {
public:
    int largestComponentSize(vector<int>& nums) {
        if (nums.empty()) return 0;

        int max_val = *max_element(nums.begin(), nums.end());
        vector<int> spf(max_val + 1, 0); // Smallest Prime Factor (SPF)

        // Compute smallest prime factors using Sieve of Eratosthenes
        for (int i = 2; i <= max_val; ++i) {
            if (spf[i] == 0) {
                for (int j = i; j <= max_val; j += i) {
                    if (spf[j] == 0) {
                        spf[j] = i;
                    }
                }
            }
        }

        // Map from prime to all indices having this prime factor
        unordered_map<int, vector<int>> primeToIndices;
        int n = nums.size();

        for (int i = 0; i < n; ++i) {
            vector<int> factors = getPrimeFactors(nums[i], spf);
            for (int factor : factors) {
                primeToIndices[factor].push_back(nums[i]);
            }
        }

        // Graph represented by adjacency list
        unordered_map<int, vector<int>> adj;
        for (const auto& [prime, indices] : primeToIndices) {
            for (int i = 1; i < indices.size(); ++i) {
                adj[indices[0]].push_back(indices[i]);
                adj[indices[i]].push_back(indices[0]);
            }
        }

        // DFS to find the size of connected components
        unordered_map<int, bool> visited;
        function<int(int)> dfs = [&](int node) -> int {
            visited[node] = true;
            int size = 1;
            for (int neighbor : adj[node]) {
                if (!visited[neighbor]) {
                    size += dfs(neighbor);
                }
            }
            return size;
        };

        int maxSize = 0;
        for (int num : nums) {
            if (!visited[num]) {
                maxSize = max(maxSize, dfs(num));
            }
        }

        return maxSize;
    }
    // Helper function to get prime factors using SPF
    vector<int> getPrimeFactors(int num, const vector<int>& spf) {
        vector<int> factors;
        while (num > 1) {
            factors.push_back(spf[num]);
            num /= spf[num];
        }
        return factors;
    }
};


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