  In formal languages and automata theory, the magic number problem can be
formulated as follows: for a given integer n, is it possible to find a number d
in the range [n,2^n] such that there is no minimal deterministic finite
automaton with d states that can be simulated by an optimal nondeterministic
finite automaton with exactly n states? If such a number d exists, it is called
magic. In this paper, we consider the magic number problem in the framework of
deterministic automata with output, which are known to characterize automatic
sequences. More precisely, we investigate magic numbers for periodic sequences
viewed as either automatic, regular, or constant-recursive.
