Many mathematicians have been interested in the study of recursive sequences. Among them, a class of “chaotic” sequences are named “meta-Fibonacci sequences.” The main example of meta-Fibonacci sequence was introduced by Hofstadter, and it is called the Q-sequence. Recently, Alkan–Fox–Aybar and the author studied the pattern induced by the connection between the Q-sequence and other known sequences. Here, we continue this program by studying a “Mertens’ version” of the Hofstadter sequence, defined (for ) by , where µ(n) is the Möbius function. In particular, as we shall see, this function encodes many interesting properties which relate prime numbers to “meta-sequences”.

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关于Hofstadter-Mertens函数的一些性质

许多数学家对递归序列的研究很感兴趣。其中，一类“混沌”序列称为“元斐波那契序列”。Hofstadter介绍了元斐波那契序列的主要例子，称为Q序列。最近，Alkan–Fox–Aybar及其作者研究了Q序列与其他已知序列之间的连接所诱导的模式。在这里，我们通过研究Hofstadter序列的“ Mertens版本”（定义为）由，其中µ（ n）是莫比乌斯函数。特别是，正如我们将看到的，该函数对许多有趣的属性进行了编码，这些属性将质数与“元序列”相关联。