Some i-MARK games

Let S be a set of positive integers, and let D be a set of integers larger than 1. The game [Formula presented] is an impartial combinatorial game introduced by Sopena (2016), which is played with a single pile of tokens. In each turn, a player can subtract s∈S from the pile, or divide the size of the pile by d∈D, if the pile size is divisible by d. Sopena partially analyzed the games with S=[1,t−1] and D={d} for d≢1(modt), but left the case d≡1(modt) open. We solve this problem by calculating the Sprague–Grundy function of [Formula presented] for d≡1(modt), for all t,d≥2. We also calculate the Sprague–Grundy function of [Formula presented] for all k, and show that it exhibits similar behavior. Finally, following Sopena's suggestion to look at games with |D|>1, we derive some partial results for the game [Formula presented], whose Sprague–Grundy function seems to behave erratically and does not show any clear pattern. We prove that each value 0,1,2 occurs infinitely often in its SG sequence, with a maximum gap length between consecutive appearances.