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

TL;DR

This paper investigates the $k$-G"obel sequence, showing that for any fixed $k \

Contribution

It proves that the $k$-G"obel sequence remains integer-valued for all initial terms up to n=18 for any $k \\geq 2$, addressing a question from manga-inspired research.

Findings

$g_{k,n}$ is always an integer for $0 \\leq n \\leq 18$ and $k \\geq 2$

Sequence behaves like an integer sequence in initial terms

Addresses a question from manga-inspired mathematical research

Abstract

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 $0 \leq n \leq 18$.