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

TL;DR

This work analyzes $k$-Göbel sequences, defined by $g_{k,0}=1$ and $g_{k,n}=\frac{1}{n}\bigl(1+\sum_{j=0}^{n-1} g_{k,j}^k\bigr)$, to determine how long they stay integral. The authors prove that $g_{k,n}$ is integral for all $k\ge2$ and $0\le n\le18$, and establish the critical property that the minimal index $N_k$ at which integrality fails is $19$, with $N_k=19$ if and only if $k\equiv6,14\pmod{18}$. Their strategy rewrites the recurrence, reduces the integrality question to $p$-adic conditions $g_{k,n}\in\mathbb{Z}_{(p)}$, and uses a finite reduction via Euler’s theorem to limit the range of $k$ that must be checked. A computational framework in Mathematica verifies these $p$-adic conditions for primes up to 17 and determines the precise $N_k$ pattern from data at $p=19$, while also discussing broader implications and open questions about the set $N$ and sensitivity to initial values.

Abstract

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 $k \geq 2$ and $0 \leq n \leq 18$.

Figures (1)

• Figure 1: The excerpt from Seisu-tan KobayashiSeki2023. They observed the list of $N_k$ and asked the above question.

Theorems & Definitions (13)

• Definition 1.1
• Theorem 1.2
• Lemma 2.1
• proof
• Example 2.2
• Example 2.3
• Example 2.4
• Definition 2.5
• Lemma 2.6
• proof