2818 - Apply Operations to Maximize Score.

Hard

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 <code>10⁹ + 7</code>.

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 nums2, ..., 2. nums2 is the only element in this subarray. Hence, we multiply the score by nums2. The score becomes 1 \* 9 = 9.

• Choose subarray nums2, ..., 3. Both nums2 and nums3 have a prime score of 1, but nums2 has the smaller index. Hence, we multiply the score by nums2. 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 nums0, ..., 0. nums0 is the only element in this subarray. Hence, we multiply the score by nums0. The score becomes 1 \* 19 = 19.

• Choose subarray nums5, ..., 5. nums5 is the only element in this subarray. Hence, we multiply the score by nums5. The score becomes 19 \* 18 = 342.

• Choose subarray nums2, ..., 3. Both nums2 and nums3 have a prime score of 2, but nums2 has the smaller index. Hence, we multipy the score by nums2. The score becomes 342 \* 14 = 4788. It can be proven that 4788 is the highest score one can obtain.

Constraints:

• <code>1 <= nums.length == n <= 10⁵</code>

• <code>1 <= numsi<= 10⁵</code>

• <code>1 <= k <= min(n * (n + 1) / 2, 10⁹)</code>