  Inspired by Episode 3 of the Japanese manga "Seisu-tan" by Doom Kobayashi and
Shin-ichiro Seki, we investigate the $k$-G\"{o}bel sequence $(g_{k,n})_n$ named
after Fritz G\"{o}bel. 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 $k \geq 2$ and $0 \leq n \leq 18$.
