<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
from functools import lru_cache
from typing import List, Tuple

PRIMES = (2, 3, 5, 7, 11, 13, 17, 19, 23, 29)

BIG_NUMBER = 10 ** 9 + 7


@lru_cache(maxsize=32)
def factor(n) -> Tuple[int, bool]:
    """
    :param n: 1 < n <= max(PRIMES)
    :return: (factors in bit mask, has duplicate primes)
    """
    output = 0

    for e in PRIMES:
        while n > 1 and n % e == 0:
            mask = 1 << e

            if mask & output:
                return -1, False

            output |= mask

            n //= e

        if n == 1:
            break

    return output, True


class Solution:
    def numberOfGoodSubsets(self, nums: List[int]) -> int:
        masks = []

        for e in nums:
            if 1 < e and (fr := factor(e))[1]:
                masks.append(fr[0])

        cnt = Counter(masks)
        good_nums = Counter({0: 0})

        for mask in cnt:
            for f in tuple(good_nums):
                if f & mask:  # some prime dividing "mask" is also dividing the "f"
                    continue

                new_mask = f | mask

                count_for_new_mask = good_nums[new_mask] + cnt[mask] * (good_nums[f] or 1)

                good_nums[new_mask] = count_for_new_mask % BIG_NUMBER

        effect_of_one = pow(2, nums.count(1), BIG_NUMBER)
        total_subsets_without_one = sum(good_nums.values()) % BIG_NUMBER

        return (effect_of_one * total_subsets_without_one) % BIG_NUMBER
1556