310507: CF1844B. Permutations & Primes
Description
You are given a positive integer $n$.
In this problem, the $\operatorname{MEX}$ of a collection of integers $c_1,c_2,\dots,c_k$ is defined as the smallest positive integer $x$ which does not occur in the collection $c$.
The primality of an array $a_1,\dots,a_n$ is defined as the number of pairs $(l,r)$ such that $1 \le l \le r \le n$ and $\operatorname{MEX}(a_l,\dots,a_r)$ is a prime number.
Find any permutation of $1,2,\dots,n$ with the maximum possible primality among all permutations of $1,2,\dots,n$.
Note:
- A prime number is a number greater than or equal to $2$ that is not divisible by any positive integer except $1$ and itself. For example, $2,5,13$ are prime numbers, but $1$ and $6$ are not prime numbers.
- A permutation of $1,2,\dots,n$ 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).
Each test contains multiple test cases. The first line contains the number of test cases $t$ ($1 \le t \le 10^4$). The description of the test cases follows.
The only line of each test case contains a single integer $n$ ($1 \le n \le 2 \cdot 10^5$).
It is guaranteed that the sum of $n$ over all test cases does not exceed $2 \cdot 10^5$.
OutputFor each test case, output $n$ integers: a permutation of $1,2,\dots,n$ that achieves the maximum possible primality.
If there are multiple solutions, print any of them.
ExampleInput3 2 1 5Output
2 1 1 5 2 1 4 3Note
In the first test case, there are $3$ pairs $(l,r)$ with $1 \le l \le r \le 2$, out of which $2$ have a prime $\operatorname{MEX}(a_l,\dots,a_r)$:
- $(l,r) = (1,1)$: $\operatorname{MEX}(2) = 1$, which is not prime.
- $(l,r) = (1,2)$: $\operatorname{MEX}(2,1) = 3$, which is prime.
- $(l,r) = (2,2)$: $\operatorname{MEX}(1) = 2$, which is prime.
In the second test case, $\operatorname{MEX}(1) = 2$ is prime, so the primality is $1$.
In the third test case, the maximum possible primality is $8$.
Input
题意翻译
你需要构造一个长度为 $n$ 的序列，使其满足以下条件： - 包含 $1\sim n$ 的所有数 - 最大化满足如下条件的 $(l,r)$ 的对数：$1\leq l\leq r\leq n$ 且 $\operatorname{MEX}(a_l,a_{l+1}\dots a_{r-1},a_r)$ 为质数 构造任意一种即可。 共 $T$ 组数据。 by @[Larryyu](https://www.luogu.com.cn/user/475329)Output
给定一个正整数n。定义一个数列的MEX为最小的正整数x，使得x不在数列中。定义一个数组的质数性为满足以下条件的(l,r)对的数量：1≤l≤r≤n，且MEX(ala,…,ar)是一个质数。求1,2,…,n的任意排列，使得其质数性在所有排列中最大。
输入输出数据格式：
输入：
第一行包含一个整数t（1≤t≤10^4），表示测试用例的数量。接下来t行，每行包含一个整数n（1≤n≤2×10^5），表示数组的大小。所有测试用例的n之和不超过2×10^5。
输出：
对于每个测试用例，输出n个整数：一个使得质数性最大的1,2,…,n的排列。如果有多个解，输出其中任意一个。
样例输入：
3
2
1
5
样例输出：
2 1
1
5 2 1 4 3题目大意： 给定一个正整数n。定义一个数列的MEX为最小的正整数x，使得x不在数列中。定义一个数组的质数性为满足以下条件的(l,r)对的数量：1≤l≤r≤n，且MEX(ala,…,ar)是一个质数。求1,2,…,n的任意排列，使得其质数性在所有排列中最大。 输入输出数据格式： 输入： 第一行包含一个整数t（1≤t≤10^4），表示测试用例的数量。接下来t行，每行包含一个整数n（1≤n≤2×10^5），表示数组的大小。所有测试用例的n之和不超过2×10^5。 输出： 对于每个测试用例，输出n个整数：一个使得质数性最大的1,2,…,n的排列。如果有多个解，输出其中任意一个。 样例输入： 3 2 1 5 样例输出： 2 1 1 5 2 1 4 3