Woeful Permutation
题意翻译
给定整数 $n$,构造一个 $1$ 到 $n$ 的排列 $p$,使得 $\sum_{i=1}^nlcm(i,p_i)$ 最大。
$lcm(x,y)$ 表示 $x$ 和 $y$ 的最小公倍数。
如果有多种答案,输出任意一种即可。
题目描述
I wonder, does the falling rain
Forever yearn for it's disdain?
Effluvium of the Mind
You are given a positive integer $ n $ .
Find any permutation $ p $ of length $ n $ such that the sum $ \operatorname{lcm}(1,p_1) + \operatorname{lcm}(2, p_2) + \ldots + \operatorname{lcm}(n, p_n) $ is as large as possible.
Here $ \operatorname{lcm}(x, y) $ denotes the [least common multiple (LCM)](https://en.wikipedia.org/wiki/Least_common_multiple) of integers $ x $ and $ y $ .
A permutation is an array consisting of $ n $ distinct integers from $ 1 $ to $ n $ in arbitrary order. For example, $ [2,3,1,5,4] $ is a permutation, but $ [1,2,2] $ is not a permutation ( $ 2 $ appears twice in the array) and $ [1,3,4] $ is also not a permutation ( $ n=3 $ but there is $ 4 $ in the array).
输入输出格式
输入格式
Each test contains multiple test cases. The first line contains the number of test cases $ t $ ( $ 1 \le t \le 1\,000 $ ). Description of the test cases follows.
The only line for each test case contains a single integer $ n $ ( $ 1 \le n \le 10^5 $ ).
It is guaranteed that the sum of $ n $ over all test cases does not exceed $ 10^5 $ .
输出格式
For each test case print $ n $ integers $ p_1 $ , $ p_2 $ , $ \ldots $ , $ p_n $ — the permutation with the maximum possible value of $ \operatorname{lcm}(1,p_1) + \operatorname{lcm}(2, p_2) + \ldots + \operatorname{lcm}(n, p_n) $ .
If there are multiple answers, print any of them.
输入输出样例
输入样例 #1
2
1
2
输出样例 #1
1
2 1
说明
For $ n = 1 $ , there is only one permutation, so the answer is $ [1] $ .
For $ n = 2 $ , there are two permutations:
- $ [1, 2] $ — the sum is $ \operatorname{lcm}(1,1) + \operatorname{lcm}(2, 2) = 1 + 2 = 3 $ .
- $ [2, 1] $ — the sum is $ \operatorname{lcm}(1,2) + \operatorname{lcm}(2, 1) = 2 + 2 = 4 $ .