<p>An <strong>ugly number</strong> is a positive integer whose prime factors are limited to <code>2</code>, <code>3</code>, and <code>5</code>.</p>

<p>Given an integer <code>n</code>, return <em>the</em> <code>n<sup>th</sup></code> <em><strong>ugly number</strong></em>.</p>

<p>&nbsp;</p>
<p><strong class="example">Example 1:</strong></p>

<pre>
<strong>Input:</strong> n = 10
<strong>Output:</strong> 12
<strong>Explanation:</strong> [1, 2, 3, 4, 5, 6, 8, 9, 10, 12] is the sequence of the first 10 ugly numbers.
</pre>

<p><strong class="example">Example 2:</strong></p>

<pre>
<strong>Input:</strong> n = 1
<strong>Output:</strong> 1
<strong>Explanation:</strong> 1 has no prime factors, therefore all of its prime factors are limited to 2, 3, and 5.
</pre>

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

<ul>
	<li><code>1 &lt;= n &lt;= 1690</code></li>
</ul>


long long my_pow(long long a, int n) {
    if (n == 0) return 1;
    if (n == 1 && a == 1) return a;
    if (n % 2 == 0) return my_pow(a * a, n / 2);
    return a * my_pow(a, n - 1);
}

class Solution {

    struct number {
        int pow2;
        int pow3;
        int pow5;

        bool operator>(const number& other) const {
            return my_pow(2, pow2) * my_pow(3, pow3) * my_pow(5, pow5) >
                my_pow(2, other.pow2) * my_pow(3, other.pow3) * my_pow(5, other.pow5);
        }
    };
public:
    int nthUglyNumber(int n) {
        unordered_set<int> s;
        priority_queue<number, vector<number>, greater<number>> q;
        number t = {0, 0, 0};
        q.push(t);
        for (int i = 1; i < n; ++i) {
            auto curr = q.top();
            q.pop();
            int num2 = my_pow(2, curr.pow2 + 1) * my_pow(3, curr.pow3) * my_pow(5, curr.pow5);
            if (s.count(num2) == 0) {
                q.push({curr.pow2 + 1, curr.pow3, curr.pow5});
                s.insert(num2);
            }
            int num3 = my_pow(2, curr.pow2) * my_pow(3, curr.pow3 + 1) * my_pow(5, curr.pow5);
            if (s.count(num3) == 0) {
                q.push({curr.pow2, curr.pow3 + 1, curr.pow5});
                s.insert(num3);
            }
            int num5 = my_pow(2, curr.pow2) * my_pow(3, curr.pow3) * my_pow(5, curr.pow5 + 1);
            if (s.count(num5) == 0) {
                q.push({curr.pow2, curr.pow3, curr.pow5 + 1});
                s.insert(num5);
            }
        }
        return my_pow(2, q.top().pow2) * my_pow(3, q.top().pow3) * my_pow(5, q.top().pow5);
    }
};
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