B. Permutations & Primes
constructive algorithms; math
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 \(8\).