<p>You are given an integer array <code>nums</code>. We call a subset of <code>nums</code> <strong>good</strong> if its product can be represented as a product of one or more <strong>distinct prime</strong> numbers.</p>

<ul>
	<li>For example, if <code>nums = [1, 2, 3, 4]</code>:

	<ul>
		<li><code>[2, 3]</code>, <code>[1, 2, 3]</code>, and <code>[1, 3]</code> are <strong>good</strong> subsets with products <code>6 = 2*3</code>, <code>6 = 2*3</code>, and <code>3 = 3</code> respectively.</li>
		<li><code>[1, 4]</code> and <code>[4]</code> are not <strong>good</strong> subsets with products <code>4 = 2*2</code> and <code>4 = 2*2</code> respectively.</li>
	</ul>
	</li>
</ul>

<p>Return <em>the number of different <strong>good</strong> subsets in </em><code>nums</code><em> <strong>modulo</strong> </em><code>10<sup>9</sup> + 7</code>.</p>

<p>A <strong>subset</strong> of <code>nums</code> is any array that can be obtained by deleting some (possibly none or all) elements from <code>nums</code>. Two subsets are different if and only if the chosen indices to delete are different.</p>

<p>&nbsp;</p>
<p><strong class="example">Example 1:</strong></p>

<pre>
<strong>Input:</strong> nums = [1,2,3,4]
<strong>Output:</strong> 6
<strong>Explanation:</strong> The good subsets are:
- [1,2]: product is 2, which is the product of distinct prime 2.
- [1,2,3]: product is 6, which is the product of distinct primes 2 and 3.
- [1,3]: product is 3, which is the product of distinct prime 3.
- [2]: product is 2, which is the product of distinct prime 2.
- [2,3]: product is 6, which is the product of distinct primes 2 and 3.
- [3]: product is 3, which is the product of distinct prime 3.
</pre>

<p><strong class="example">Example 2:</strong></p>

<pre>
<strong>Input:</strong> nums = [4,2,3,15]
<strong>Output:</strong> 5
<strong>Explanation:</strong> The good subsets are:
- [2]: product is 2, which is the product of distinct prime 2.
- [2,3]: product is 6, which is the product of distinct primes 2 and 3.
- [2,15]: product is 30, which is the product of distinct primes 2, 3, and 5.
- [3]: product is 3, which is the product of distinct prime 3.
- [15]: product is 15, which is the product of distinct primes 3 and 5.
</pre>

<p>&nbsp;</p>
<p><strong>Constraints:</strong></p>

<ul>
	<li><code>1 &lt;= nums.length &lt;= 10<sup>5</sup></code></li>
	<li><code>1 &lt;= nums[i] &lt;= 30</code></li>
</ul>

class Solution:
    def numberOfGoodSubsets(self, nums):
        mod = 10**9 + 7
        primes = [2, 3, 5, 7, 11, 13, 17, 19, 23, 29]
        freq = [0] * 31
        for num in nums:
            freq[num] += 1
        
        # dp[i] represents the number of ways to form the product using primes represented by the bitmask i
        dp = [0] * (1 << len(primes))
        dp[0] = 1
        
        for i in range(2, 31):
            if freq[i] == 0:
                continue
            
            # Check if i is a valid number (not a product of non-distinct primes)
            mask = 0
            for j, prime in enumerate(primes):
                if i % prime == 0 and i % (prime * prime) == 0:
                    break
                if i % prime == 0:
                    mask |= (1 << j)
            else:
                for j in range((1 << len(primes)) - 1, -1, -1):
                    if (j & mask) == mask:
                        dp[j] = (dp[j] + dp[j ^ mask] * freq[i]) % mod
        
        # Exclude the empty subset and count only the subsets that use at least one prime
        result = sum(dp[1:]) % mod
        
        # Account for the number of ways to choose the number 1 (which doesn't affect the product)
        result = (result * pow(2, freq[1], mod)) % mod
        
        return result

# Example usage:
solution = Solution()
nums = [1, 2, 3, 4]
print(solution.numberOfGoodSubsets(nums))  # Output: 6

nums = [4, 2, 3, 15]
print(solution.numberOfGoodSubsets(nums))  # Output: 5
20800