Maximize Number of Nice Divisors

Problem Description

You are given a positive integer primeFactors. You are asked to construct a positive integer n that satisfies the following conditions:

• The number of prime factors of n (not necessarily distinct) is at most primeFactors.
• The number of nice divisors of n is maximized. A divisor of n is nice if it is divisible by every prime factor of n.

Return the number of nice divisors of n. Since that number can be too large, return it modulo 10^9 + 7.

Key Insights

• The problem revolves around maximizing the number of nice divisors by strategically distributing the prime factors.
• The optimal distribution of prime factors tends to be balanced, meaning using similar sizes of prime factors leads to more divisors.
• The number of divisors can be calculated based on the prime factorization of a number where if n = p1^e1 * p2^e2 * ... * pk^ek, the number of divisors is given by (e1 + 1) * (e2 + 1) * ... * (ek + 1).

Space and Time Complexity

Time Complexity: O(log(primeFactors))
Space Complexity: O(1)

Solution

To solve this problem, we can utilize the concept of distributing the prime factors into groups to maximize the product. The optimal way to achieve this is to keep the groups as balanced as possible, leading to a higher product of divisors.

1. Calculate k, the number of groups we can form with primeFactors.
2. Use the formula for the number of divisors based on the distribution of prime factors.
3. Return the result modulo 10^9 + 7.

Code Solutions