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,2n] 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.

翻译：在形式语言与自动机理论中，幻数问题可表述为：对于给定整数n，是否能在区间[n,2n]内找到一个整数d，使得不存在一个具有d个状态的最小确定型有限自动机，能够被恰好具有n个状态的最优非确定型有限自动机模拟？若存在这样的数d，则称其为幻数。本文在带输出的确定型自动机框架下考虑幻数问题，这类自动机以刻画自动序列而著称。具体而言，我们研究了周期序列的幻数，这些周期序列可视为自动序列、正则序列或常递归序列。