2818. Apply Operations to Maximize Score ¶

Problem

You are given an array nums of n positive integers and an integer k.

Initially, you start with a score of 1. You have to maximize your score by applying the following operation at most k times:

Choose any non-empty subarray nums[l, ..., r] that you haven't chosen previously.
Choose an element x of nums[l, ..., r] with the highest prime score. If multiple such elements exist, choose the one with the smallest index.
Multiply your score by x.

Here, nums[l, ..., r] denotes the subarray of nums starting at index l and ending at the index r, both ends being inclusive.

The prime score of an integer x is equal to the number of distinct prime factors of x. For example, the prime score of 300 is 3 since 300 = 2 * 2 * 3 * 5 * 5.

Return the maximum possible score after applying at most k operations.

Since the answer may be large, return it modulo 10⁹+ 7.

Example 1:

Input: nums = [8,3,9,3,8], k = 2
Output: 81
Explanation: To get a score of 81, we can apply the following operations:
- Choose subarray nums[2, ..., 2]. nums[2] is the only element in this subarray. Hence, we multiply the score by nums[2]. The score becomes 1 * 9 = 9.
- Choose subarray nums[2, ..., 3]. Both nums[2] and nums[3] have a prime score of 1, but nums[2] has the smaller index. Hence, we multiply the score by nums[2]. The score becomes 9 * 9 = 81.
It can be proven that 81 is the highest score one can obtain.

Example 2:

Input: nums = [19,12,14,6,10,18], k = 3
Output: 4788
Explanation: To get a score of 4788, we can apply the following operations: 
- Choose subarray nums[0, ..., 0]. nums[0] is the only element in this subarray. Hence, we multiply the score by nums[0]. The score becomes 1 * 19 = 19.
- Choose subarray nums[5, ..., 5]. nums[5] is the only element in this subarray. Hence, we multiply the score by nums[5]. The score becomes 19 * 18 = 342.
- Choose subarray nums[2, ..., 3]. Both nums[2] and nums[3] have a prime score of 2, but nums[2] has the smaller index. Hence, we multipy the score by nums[2]. The score becomes 342 * 14 = 4788.
It can be proven that 4788 is the highest score one can obtain.

Constraints:

1 <= nums.length == n <= 10⁵
1 <= nums[i] <= 10⁵
1 <= k <= min(n * (n + 1) / 2, 10⁹)