Problem Statement
Prime Factor Sequence
You're working on a mathematical sequence generator that produces numbers with specific prime factorization properties.
An ugly number is a positive integer whose prime factors are limited to 2, 3, and 5.
Given an integer n, return the nth ugly number in the sequence, where 1 is the first ugly number.
Examples
Example 1:
Input: n = 10
Output: 12
Explanation: [1, 2, 3, 4, 5, 6, 8, 9, 10, 12] is the sequence of the first 10 ugly numbers.
Example 2:
Input: n = 1
Output: 1
Explanation: 1 has no prime factors, therefore all of its prime factors are limited to 2, 3, and 5.
Constraints
- 1 <= n <= 1690
Problem Breakdown
To solve this problem, we need to:
- Every ugly number (except 1) can be generated by multiplying a previous ugly number by either 2, 3, or 5
- We can use dynamic programming to generate the sequence of ugly numbers
- To avoid duplicates, we need to keep track of which previous ugly numbers have been multiplied by 2, 3, and 5
- The key insight is to maintain three pointers to track which previous ugly numbers to multiply by 2, 3, and 5
- A min heap can also be used to efficiently find the next ugly number in the sequence
- The problem can be solved in O(n) time and O(n) space
String Problems
Learning Objective
Apply string manipulation concepts to solve a real-world problem.
Prime Factor Sequence
You're working on a mathematical sequence generator that produces numbers with specific prime factorization properties.
An ugly number is a positive integer whose prime factors are limited to 2, 3, and 5.
Given an integer n, return the nth ugly number in the sequence, where 1 is the first ugly number.
Examples
Example 1:
[1, 2, 3, 4, 5, 6, 8, 9, 10, 12] is the sequence of the first 10 ugly numbers.
Example 2:
1 has no prime factors, therefore all of its prime factors are limited to 2, 3, and 5.
Problem Breakdown
Constraints:
- 1 <= n <= 1690
Key Insights:
Every ugly number (except 1) can be generated by multiplying a previous ugly number by either 2, 3, or 5
We can use dynamic programming to generate the sequence of ugly numbers
To avoid duplicates, we need to keep track of which previous ugly numbers have been multiplied by 2, 3, and 5
The key insight is to maintain three pointers to track which previous ugly numbers to multiply by 2, 3, and 5
A min heap can also be used to efficiently find the next ugly number in the sequence
The problem can be solved in O(n) time and O(n) space
Real-World Application
This problem has several practical applications:
Number Theory
Studying sequences with specific prime factorization properties in mathematical research.
Sequence Generation
Generating sequences with specific properties for cryptographic applications.
Probability Models
Modeling systems where only certain factors contribute to outcomes.
Financial Modeling
Creating models where growth is limited to specific multiplicative factors.
Algorithm Design
Studying efficient ways to generate sequences with specific constraints.