<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>

using ll = long long;

class Solution {
public:
    int nthUglyNumber(int n) {
        ll max_k = 2 * (n / 3ll + 1) * n * n;

        std::vector<ll> nums;
        for (ll k5 = 1; k5 <= max_k; k5 *= 5) {
            for (ll k3 = 1; k3 <= max_k / k5; k3 *= 3) {
                for (ll k2 = 1; k2 <= max_k / (k3 * k5); k2 *= 2) {
                    nums.push_back(k2 * k3 * k5);
                }
            }
        }

        std::sort(nums.begin(), nums.end());
        return nums[n - 1];
    }
};
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