Problem Description
You are given two integers n and maxValue, which are used to describe an ideal array. An array arr of length n is considered ideal if every arr[i] is a value from 1 to maxValue, and every arr[i] is divisible by arr[i - 1]. Return the number of distinct ideal arrays of length n, modulo 10^9 + 7.
Key Insights
- Each element in the array must be divisible by the previous element.
- Arrays can start with any number from 1 to maxValue.
- The number of distinct ideal arrays can be computed using combinatorial counting based on divisors.
Space and Time Complexity
Time Complexity: O(n * log(maxValue)) Space Complexity: O(maxValue)
Solution
To solve the problem, we can utilize dynamic programming combined with number theory. We will maintain a count of how many ideal arrays can be formed starting with each number from 1 to maxValue. For each number, we will consider all its multiples up to maxValue and accumulate the counts from the previous step, thus forming valid ideal arrays. The final result will be the sum of all counts for arrays of length n, taken modulo 10^9 + 7.