Two Games on Arithmetic Functions: SALIQUANT and NONTOTIENT

TL;DR

This paper studies two arithmetic function-based impartial games, analyzing their Sprague-Grundy sequences, establishing key lemmas, conjectures, and partial calculations to understand their combinatorial and number-theoretic properties.

Contribution

It introduces new number-theoretic lemmas, a fundamental theorem, and conjectures for the Sprague-Grundy sequences of these two novel arithmetic games.

Findings

Derived number-theoretic lemmas for the games

Formulated conjectures on the density of Sprague-Grundy values

Calculated specific Sprague-Grundy values and identified relevant class functions

Abstract

We investigate the Sprague-Grundy sequences for two normal-play impartial games based on arithmetic functions, first described by Iannucci and Larsson in \cite{sum}. In each game, the set of positions is N (natural numbers). In saliquant, the options are to subtract a non-divisor. Here we obtain several nice number theoretic lemmas, a fundamental theorem, and two conjectures about the eventual density of Sprague-Grundy values. In nontotient, the only option is to subtract the number of relatively prime residues. Here are able to calculate certain Sprague-Grundy values, and start to understand an appropriate class function.