How long can $k$-Göbel sequences remain integers?

Inspired by Episode 3 of the Japanese manga “Seisu-tan” by Doom Kobayashi and Shin-ichiro Seki, we investigate the $k$-Göbel sequence $(g_{k,n}{)}_n$ named after Fritz Göbel. Although the sequence is generally defined as rational, quite a few initial terms behave like an integer sequence. This article addresses a question raised in Seisu-tan and shows that $g_{k,n}$ is always an integer for any $k\ge 2$ and $0\le n\le 18$.