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Inspired by Episode 3 of the Japanese manga "Seisu-tan" by Doom Kobayashi andShin-ichiro Seki, we investigate the $k$-G\"{o}bel sequence $(g_{k,n})_n$ namedafter Fritz G\"{o}bel. Although the sequence is generally defined as rational,quite a few initial terms behave like an integer sequence. This articleaddresses a question raised in Seisu-tan and shows that $g_{k,n}$ is always aninteger for any $k \geq 2$ and $0 \leq n \leq 18$.

How long can $k$-G\"{o}bel sequences remain integers?