<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>

from collections import Counter
class Solution:
    def nthUglyNumber(self, n: int) -> int:
        a = [1,2,3]
        if  n < 3:
            return a[n-1]
        last_start = -1
        for _ in range(3, n):
            prev = a[-1]*2
            for i_index in range(last_start+1, len(a)):
                i = a[i_index]
                if i*5 < a[-1]:
                    last_start = i_index
                if i * 2 > prev:
                    break
                for j in [2,3,5]:
                    if i*j > a[-1]:
                        prev = min(prev, i*j)
                        break
            a.append(prev)
        return a[-1]
16200