D. Turtle and Multiplication
constructive algorithms; dfs and similar; graphs; number theory
Turtle just learned how to multiply two integers in his math class, and he was very excited.Then Piggy gave him an integer \(n\), and asked him to construct a sequence \(a_1, a_2, \ldots, a_n\) consisting of integers which satisfied the following conditions: For all \(1 \le i \le n\), \(1 \le a_i \le 3 \cdot 10^5\). For all \(1 \le i < j \le n - 1\), \(a_i \cdot a_{i + 1} \ne a_j \cdot a_{j + 1}\). Of all such sequences, Piggy asked Turtle to find the one with the minimum number of distinct elements.Turtle definitely could not solve the problem, so please help him!
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 first line of each test case contains a single integer \(n\) (\(2 \le n \le 10^6\)) — the length of the sequence \(a\).It is guaranteed that the sum of \(n\) over all test cases does not exceed \(10^6\).
For each test case, output \(n\) integers \(a_1, a_2, \ldots, a_n\) — the elements of the sequence \(a\).If there are multiple answers, print any of them.
In the third test case, \(a = [3, 4, 2, 6]\) violates the second condition since \(a_1 \cdot a_2 = a_3 \cdot a_4\). \(a = [2, 3, 4, 4]\) satisfy the conditions but its number of distinct elements isn't minimum.