Effective MC-finiteness

TL;DR

This paper investigates the computability of the function mapping pairs of integers to sequence values modulo m for MC-finite sequences, revealing cases where this function is effectively computable.

Contribution

It extends previous work by analyzing when the modulo sequence function for MC-finite sequences can be computed effectively.

Findings

Identifies conditions under which the modulo sequence function is computable.

Provides examples of MC-finite sequences with effectively computable modulo functions.

Contrasts with prior results showing some MC-finite sequences have non-computable functions.

Abstract

An integer sequence $(a_n)_{n \in \mathbb{N}}$ is \emph{MC-finite} if for all $m$, the sequence $a_n \bmod m$ is eventually periodic. There are MC-finite sequences $(a_n)_{n \in \mathbb{N}}$ such that the function $F: (m,n) \mapsto a_n \bmod m$ is not computable. In \cite{filmus2023mc} we presented concrete examples of MC-finite sequences taken from the Online Encyclopedia of Integer Sequences (OEIS) without discussing the computability of $F$. In this paper we discuss cases when this $F$ is effectively computable.