We consider one-dimensional linear cellular automata whose states are the integers modulo a prime power P~d and their orbital patterns. We are particulary interested in initial states and their orbital patterns which are invariant under a certain coarse-graining operation. We show that these coarse-graining invariant initial states are p-automatic. The relationship between the solutions of a certain family of coarse-graining invariant problems concerning linear cellular automata over the integers modulo p~n is investigated.
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