How Long Can <i>k</i> -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 (gk,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 gk,n is always an integer for any k≥2 and 0≤n≤18.