Two games on arithmetic functions: SALIQUANT and NONTOTIENT

Abstract

We investigate the Sprague-Grundy sequences for two normal-play impartial games based on arithmetic functions, first described by Iannucci and Larsson in a book chapter. In each game, the set of positions is N. 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 we are able to calculate certain Sprague-Grundy values and start to understand an appropriate class function.