A15525. Range = √Sum

You are given an integer $n$ . Find a sequence of $n$ distinct integers $a_1, a_2, \dots, a_n$ such that $1 \leq a_i \leq 10^9$ for all $i$ and $$$$\max(a_1, a_2, \dots, a_n) - \min(a_1, a_2, \dots, a_n)= \sqrt{a_1 + a_2 + \dots + a_n}. $$$$

It can be proven that there exists a sequence of distinct integers that satisfies all the conditions above.

The first line of input contains $t$ ( $1 \leq t \leq 10^4$ ) — the number of test cases.

The first and only line of each test case contains one integer $n$ ( $2 \leq n \leq 3 \cdot 10^5$ ) — the length of the sequence you have to find.

The sum of $n$ over all test cases does not exceed $3 \cdot 10^5$ .

For each test case, output $n$ space-separated distinct integers $a_1, a_2, \dots, a_n$ satisfying the conditions in the statement.

If there are several possible answers, you can output any of them. Please remember that your integers must be distinct!

In the first test case, the maximum is $3$ , the minimum is $1$ , the sum is $4$ , and $3 - 1 = \sqrt{4}$ .

In the second test case, the maximum is $29$ , the minimum is $18$ , the sum is $121$ , and $29-18 = \sqrt{121}$ .

For each test case, the integers are all distinct.