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 Image 1 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.