A16006. Permutations & Primes

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 $2 \cdot 10^5$ .

For 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.

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.

Therefore, the primality is $2$ .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 $8$ .