CF2060F Multiplicative Arrays

You're given integers $ k $ and $ n $ . For each integer $ x $ from $ 1 $ to $ k $ , count the number of integer arrays $ a $ such that all of the following are satisfied: - $ 1 \leq |a| \leq n $ where $ |a| $ represents the length of $ a $ . - $ 1 \leq a_i \leq k $ for all $ 1 \leq i \leq |a| $ . - $ a_1 \times a_2 \times \dots \times a_{|a|}=x $ (i.e., the product of all elements is $ x $ ). Note that two arrays $ b $ and $ c $ are different if either their lengths are different, or if there exists an index $ 1 \leq i \leq |b| $ such that $ b_i\neq c_i $ . Output the answer modulo $ 998\,244\,353 $ .

The first line contains an integer $ t $ ( $ 1 \leq t\leq 10^3 $ ) — the number of independent test cases. The only line of each test case contains two integers $ k $ and $ n $ ( $ 1 \leq k \leq 10^5, 1\leq n \leq 9\cdot 10^8 $ ). It is guaranteed that the sum of $ k $ over all test cases does not exceed $ 10^5 $ .

For each test case, output $ k $ space-separated integers on a new line: the number of arrays for $ x=1,2,\ldots,k $ , modulo $ 998\,244\,353 $ .

In the first test case, there are $ 2 $ arrays $ a $ with $ |a|\leq 2 $ and the product of elements equal to $ 1 $ : - $ [1] $ - $ [1,1] $ There are $ 3 $ arrays $ a $ with $ |a|\leq 2 $ and the product of elements equal to $ 2 $ : - $ [2] $ - $ [1,2] $ - $ [2,1] $