<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>

class Solution {
  using Int = std::int64_t;
public:
  int nthUglyNumber(int const n) {
    std::priority_queue<Int> ugly_numbers;
    std::queue<std::pair<Int, char>> q;
    q.emplace(1, '2');
    while (!q.empty()) {
      auto const [u, tag] = q.front(); q.pop();
      ugly_numbers.push(u);
      if (ugly_numbers.size() > n) ugly_numbers.pop();
      if (ugly_numbers.size() < n || u < ugly_numbers.top()) {
        if (tag == '2') q.emplace(2*u, '2');
        if (tag <= '3') q.emplace(3*u, '3');
                        q.emplace(5*u, '5');
      }
    }
    assert( ugly_numbers.size() == n );
    return static_cast<int>(ugly_numbers.top());
  }
};
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