<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 {
public:
    int nthUglyNumber(int n) {
        std::unordered_map<unsigned, bool> exists;
        std::vector<unsigned> dp(n);
        dp[0] = 1;

        int index_2 = 0;
        int index_3 = 0;
        int index_5 = 0;

        auto update_index = [&exists, &dp](int& index, int i, int multiplier) {
            while (index < i && exists[dp[index] * multiplier]) {
                index++;
            }
        };

        auto update_indexes = [&](int i) {
            update_index(index_2, i, 2);
            update_index(index_3, i, 3);
            update_index(index_5, i, 5);
        };

        for (int i = 1; i < n; i++) {
            update_indexes(i);
            unsigned ugly_2 = dp[index_2] * 2;
            unsigned ugly_3 = dp[index_3] * 3;
            unsigned ugly_5 = dp[index_5] * 5;

            if (ugly_2 < ugly_3 && ugly_2 < ugly_5) {
                dp[i] = ugly_2;
                index_2++;
            } else if (ugly_3 < ugly_5) {
                dp[i] = ugly_3;
                index_3++;
            } else {
                dp[i] = ugly_5;
                index_5++;
            }
            exists[dp[i]] = true;
        }

        return dp[n - 1];
    }
};
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