Divisibility properties of sporadic Ap\'ery-like numbers

TL;DR

This paper proves Lucas-type divisibility congruences for all sporadic Apéry-like sequences, extending Gessel's 1982 results, and explores their periodicity and prime divisibility properties.

Contribution

It establishes new divisibility congruences for all sporadic Apéry-like sequences, including cases requiring refined analysis, and studies their periodicity and prime divisibility.

Findings

Lucas congruences hold for all sporadic Apéry-like sequences.

The Almkvist–Zudilin numbers are periodic modulo 8.

Identifies primes that do not divide any term of these sequences.

Abstract

In 1982, Gessel showed that the Ap\'ery numbers associated to the irrationality of $\zeta(3)$ satisfy Lucas congruences. Our main result is to prove corresponding congruences for all sporadic Ap\'ery-like sequences. In several cases, we are able to employ approaches due to McIntosh, Samol--van Straten and Rowland--Yassawi to establish these congruences. However, for the sequences often labeled $s_{18}$ and $(\eta)$ we require a finer analysis. As an application, we investigate modulo which numbers these sequences are periodic. In particular, we show that the Almkvist--Zudilin numbers are periodic modulo $8$, a special property which they share with the Ap\'ery numbers. We also investigate primes which do not divide any term of a given Ap\'ery-like sequence.