Apply operations to maximize score leetcode 2818 python

okay, let's break down the leetcode problem 2818, "apply operations to maximize score," with a comprehensive python solution and explanation.

**problem statement (leetcode 2818)**

you are given two 0-indexed integer arrays `nums1` and `nums2` of length `n`. you are also given a positive integer `k`.

you must perform the following operation exactly `k` times:

1. choose an index `i` such that `0 = i n`.
2. increase `nums1[i]` by `nums2[i]`.

your task is to maximize the sum of the elements of `nums1` after performing the operation exactly `k` times.

return *the maximum possible sum* of the elements of `nums1` after applying the operation exactly `k` times. since the answer may be large, return it modulo `10^9 + 7`.

**constraints:**

* `1 = n = 10^5`
* `1 = k = 10^5`
* `1 = nums1[i], nums2[i] = 10^5`

**core idea and solution strategy**

the key insight here is that you want to choose the indices that yield the largest increase in `nums1` when `nums2` is added to them. this means prioritizing the indices with the largest corresponding values in `nums2`. however, there's a crucial added layer of complexity: *prime factors.* we need to consider the number of *distinct* prime factors in each `nums1[i]` because this contributes to a score multiplier.

here's the breakdown of the optimal approach:

1. **precompute prime factors:** efficiently calculate the number of distinct prime factors for each number within the constraint range (up to `10^5` in this case). we'll use the sieve of eratosthenes for this.

2. **prioritize with a heap:** use a max heap (priority queue) to store tuples: `(prime_factor_count, nums2[i], nums1[i])`. we'll prioritize based on:
* primary key: the number of distinct prime factors of `nums1[i]` (larger is better).
* secondary key (tie-breaker): `nums2[i]` (larger is better).
* tertiary key: `nums1[i]` (this is only for retaining the original value, not for sorting!)

3. **apply operations:** extract the e ...

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