D._Small_GCD题解(codeforces 911 div.2)

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已于 2023-11-29 19:41:34 修改 · 1.3k 阅读

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于 2023-11-27 17:56:46 首次发布

博客围绕数组中所有三元组较小两个数的gcd求和问题展开。先分析暴力解法复杂度高会超时，接着讲述错误思路及修正尝试。还介绍了从排行榜看到的代码思路，以及欧拉反演和容斥两种补充思路，最后指出两种补充思路本质相同。

题干：

Let $a$ , $b$ , and $c$ be integers. We define function $f (a, b, c)$ as follows:

Order the numbers $a$ , $b$ , $c$ in such a way that $\le b \le c$ . Then return $\gcd(a, b)$ , where $\gcd(a, b)$ denotes the greatest common divisor (GCD) of integers $a$ and $b$ .

So basically, we take the $\gcd$ of the $2$ smaller values and ignore the biggest one.

You are given an array $a$ of $n$ elements. Compute the sum of $f(a_i, a_j, a_k)$ for each $i$ , $j$ , $k$ , such that $\le i < j < k \le n$ .

More formally, compute

$\sum_{i = 1}^n \sum_{j = i+1}^n \sum_{k =j +1}^n f(a_i, a_j, a_k).$

输入要求：

Input

Each test contains multiple test cases. The first line contains the number of test cases $t$ ( $\le t \le 10$ ). The description of the test cases follows.

The first line of each test case contains a single integer $n$ ( $\le n \le 8 \cdot 10^4$ ) — length of the array $a$ .

The second line of each test case contains $n$ integers, $a_1, a_2, \ldots, a_n$

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