F. Multiplicative Arrays
combinatorics; dp; number theory
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 \(2\): \([2]\) \([1,2]\) \([2,1]\)