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The Number of Good Subsets

Difficulty: Hard

Problem Description

You are given an integer array nums. We call a subset of nums good if its product can be represented as a product of one or more distinct prime numbers. Return the number of different good subsets in nums modulo 10^9 + 7.

Key Insights

  • A subset is considered good if it contains only distinct prime factors in its product.
  • The numbers in nums range from 1 to 30, which allows us to utilize the properties of prime factorization effectively.
  • The number 1 does not contribute to the product of distinct primes, so it can be included in any subset without affecting the "goodness".
  • We need to consider the frequency of each number in nums since duplicate primes can affect the count of good subsets.

Space and Time Complexity

Time Complexity: O(n + m) where n is the length of nums and m is the number of distinct primes up to the maximum value in nums (30).

Space Complexity: O(m) where m is the number of distinct primes up to 30.

Solution

To solve the problem, we can follow these steps:

  1. Count Frequencies: First, count the frequency of each number in nums.
  2. Identify Distinct Primes: Identify all distinct primes up to 30 since these will be our candidates for forming good subsets.
  3. Subset Calculation: For each distinct prime, calculate the number of good subsets that can be formed using its multiples based on their frequencies.
  4. Combine Results: Combine the results from all distinct primes while considering subsets formed by including the number 1, which can be included in any combination.

This approach utilizes combinatorial counting, taking advantage of the limited range of numbers and their prime factorization properties.

Code Solutions

def count_good_subsets(nums):
    MOD = 10**9 + 7
    freq = [0] * 31
    for num in nums:
        freq[num] += 1

    # List of primes up to 30
    primes = [2, 3, 5, 7, 11, 13, 17, 19, 23, 29]
    good_count = 1  # Start with the empty subset

    for prime in primes:
        if freq[prime] > 0:
            # Each number can either be included or not included
            good_count *= (pow(2, freq[prime], MOD) - 1)
            good_count %= MOD

    # Include subsets with 1
    if freq[1] > 0:
        good_count *= pow(2, freq[1], MOD)
        good_count %= MOD

    return good_count

# Example usage
print(count_good_subsets([1, 2, 3, 4]))  # Output: 6
print(count_good_subsets([4, 2, 3, 15])) # Output: 5
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