P9308 "DTOI-5" #1f1e33

 In the middle of night.

Define the function $f(n) = \displaystyle\sum_{i = 1}^n \sum_{j = 1}^n \sum_{k = 1}^n [i + j + k = n] \operatorname{lcm}(i, \gcd(j, k))$. Given $n$, for all $1 \leq i \leq n$, output all values of $f(i) \bmod 998244353$.

One line containing one integer $n$.

One line containing $n$ integers, representing all values of $f(i) \bmod 998244353$.

**Constraints** $$ \def\or{\operatorname{or}} %\def\arrayscretch{1.5} \def\arraystretch{1.5} \begin{array}{|c|c|c|}\hline \textbf{Test Case ID}&n= &\textbf{Points}\cr\hline \sf1&100&10 \operatorname{pts}\cr\hline \sf2&10^3&10 \operatorname{pts}\cr\hline \sf3&10^4&20 \operatorname{pts}\cr\hline \sf4&10^5&20 \operatorname{pts}\cr\hline \sf5&/&40 \operatorname{pts}\cr\hline \end{array} $$ For $100\%$ of the testdata, $1 \leq n \leq 10^6$. Translated by ChatGPT 5