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g1901_2000.s1994_the_number_of_good_subsets.readme.md Maven / Gradle / Ivy

1994\. The Number of Good Subsets

Hard

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.

*   For example, if `nums = [1, 2, 3, 4]`:
    *   `[2, 3]`, `[1, 2, 3]`, and `[1, 3]` are **good** subsets with products `6 = 2*3`, `6 = 2*3`, and `3 = 3` respectively.
    *   `[1, 4]` and `[4]` are not **good** subsets with products `4 = 2*2` and `4 = 2*2` respectively.

Return _the number of different **good** subsets in_ `nums` _**modulo**_ 109 + 7.

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

**Example 1:**

**Input:** nums = [1,2,3,4]

**Output:** 6

**Explanation:** 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. 

**Example 2:**

**Input:** nums = [4,2,3,15]

**Output:** 5

**Explanation:** 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. 

**Constraints:**

*   1 <= nums.length <= 105
*   `1 <= nums[i] <= 30`