CF1937A Shuffle Party

You are given an array $ a_1, a_2, \ldots, a_n $ . Initially, $ a_i=i $ for each $ 1 \le i \le n $ . The operation $ \texttt{swap}(k) $ for an integer $ k \ge 2 $ is defined as follows: - Let $ d $ be the largest divisor $ ^\dagger $ of $ k $ which is not equal to $ k $ itself. Then swap the elements $ a_d $ and $ a_k $ . Suppose you perform $ \texttt{swap}(i) $ for each $ i=2,3,\ldots, n $ in this exact order. Find the position of $ 1 $ in the resulting array. In other words, find such $ j $ that $ a_j = 1 $ after performing these operations. $ ^\dagger $ An integer $ x $ is a divisor of $ y $ if there exists an integer $ z $ such that $ y = x \cdot z $ .

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 one integer $ n $ ( $ 1 \le n \le 10^9 $ ) — the length of the array $ a $ .

For each test case, output the position of $ 1 $ in the resulting array.

In the first test case, the array is $ [1] $ and there are no operations performed. In the second test case, $ a $ changes as follows: - Initially, $ a $ is $ [1,2,3,4] $ . - After performing $ \texttt{swap}(2) $ , $ a $ changes to $ [\underline{2},\underline{1},3,4] $ (the elements being swapped are underlined). - After performing $ \texttt{swap}(3) $ , $ a $ changes to $ [\underline{3},1,\underline{2},4] $ . - After performing $ \texttt{swap}(4) $ , $ a $ changes to $ [3,\underline{4},2,\underline{1}] $ . Finally, the element $ 1 $ lies on index $ 4 $ (that is, $ a_4 = 1 $ ). Thus, the answer is $ 4 $ .