<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>

from collections import Counter

class Solution:
    def count_dp(self, dp, m, root_primes, cnt_, root):
        tot_count = 0
        for i in dp[root]:
            if not m[i].intersection(root_primes):
                tot_count += (cnt_[i] * (sum(cnt_[j] for j in dp[i] if not root_primes.union(m[i]).intersection(m[j])) + self.count_dp(dp,m,root_primes.union(m[i]), cnt_, i))) % (10**9 + 7)
        return tot_count % (10**9 + 7)

    def count_primes(self, nums, primes, m):
        bad = {4,8,9,12,16,18,20,24,25,27,28}
        tot_count = 0
        cnt_ = Counter(nums)
        for i in bad:
            cnt_.pop(i, None)
        dp = {}
        for i in range(2,31):
            if i in cnt_ and i not in bad :
                dp[i] = sorted([
                    key for key in cnt_ if ((key not in (i,1)) and (key not in dp) and (key not in bad) and (not m[i].intersection(m[key])))
                ])
        count = 0
        for i in range(2,31):
            if i in cnt_ and i not in bad :
                c = (cnt_[i] * (sum(cnt_[j] for j in dp[i]) + self.count_dp(dp, m, m[i], cnt_, i))  % (10**9 + 7))
                count += c
        ones = cnt_.pop(1, 0)
        count += sum(cnt_.values())

        return (count * pow(2,ones,10**9 + 7)) % (10**9 + 7)

        
    def numberOfGoodSubsets(self, nums: List[int]) -> int:
        primes = {2,3,5,7,11,13,17,19,23,29}
        m = {
            i: set(j for j in primes if i%j==0) for i in range(1, 31)
        }
        return self.count_primes(nums, primes, m)
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