On the last nonzero digits of $n!$ in a given base

TL;DR

This paper investigates the sequence of last nonzero digits of factorials in various bases, characterizing when these sequences are automatic and providing methods to generate them, along with frequency analysis of digit patterns.

Contribution

It determines the conditions under which the sequence of last nonzero digits of factorials is automatic and introduces a uniform morphism for their generation.

Findings

Identifies bases for which the sequence is automatic

Provides a method to generate the sequence using uniform morphism

Analyzes the frequency of digit patterns in the sequence

Abstract

In this paper we study the sequence of strings of $k$ last nonzero digits of $n!$ in a given base $b$. We determine for which $b$ this sequence is automatic and show how to generate it using a uniform morphism. We also compute how often each possible string of $k$ digits appears as the $k$ last nonzero digits of $n!$.