On the length over which $k$-G\"obel sequences remain integers

TL;DR

This paper proves that the minimal length over which the $k$-G"obel sequence remains integral is unbounded, indicating the sequence's complexity and the limits of its integrality properties.

Contribution

It establishes that the sequence of minimal lengths for $k$-G"obel sequences to lose integrality is unbounded, a new theoretical insight.

Findings

The sequence $(N_k)_k$ is unbounded.

$k$-G"obel sequences can have arbitrarily long integer terms.

The result advances understanding of the integrality properties of these sequences.

Abstract

We prove that the sequence $(N_k)_k$, where each $N_k$ is defined as the smallest positive integer $n$ for which the $n$th term $g_{k,n}$ of the $k$-G\"obel sequence is not an integer, is unbounded.