On Some Properties of the Hofstadter–Mertens Function
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...
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| Format: | Article |
| Language: | English |
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Wiley
2020-01-01
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| Series: | Complexity |
| Online Access: | http://dx.doi.org/10.1155/2020/1816756 |
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| author | Pavel Trojovský |
| author_facet | Pavel Trojovský |
| author_sort | Pavel Trojovský |
| collection | DOAJ |
| description | 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 x>0) by x↦∑n≤xμnQn, 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”. |
| format | Article |
| id | doaj-art-ca43ed748a0a4c76bac7e3dceb62dd4f |
| institution | Kabale University |
| issn | 1076-2787 1099-0526 |
| language | English |
| publishDate | 2020-01-01 |
| publisher | Wiley |
| record_format | Article |
| series | Complexity |
| spelling | doaj-art-ca43ed748a0a4c76bac7e3dceb62dd4f2025-02-03T06:05:12ZengWileyComplexity1076-27871099-05262020-01-01202010.1155/2020/18167561816756On Some Properties of the Hofstadter–Mertens FunctionPavel Trojovský0Department of Mathematics, Faculty of Science, University of Hradec Králové, Rokitanského 62, Hradec Králové, Czech RepublicMany 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 x>0) by x↦∑n≤xμnQn, 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”.http://dx.doi.org/10.1155/2020/1816756 |
| spellingShingle | Pavel Trojovský On Some Properties of the Hofstadter–Mertens Function Complexity |
| title | On Some Properties of the Hofstadter–Mertens Function |
| title_full | On Some Properties of the Hofstadter–Mertens Function |
| title_fullStr | On Some Properties of the Hofstadter–Mertens Function |
| title_full_unstemmed | On Some Properties of the Hofstadter–Mertens Function |
| title_short | On Some Properties of the Hofstadter–Mertens Function |
| title_sort | on some properties of the hofstadter mertens function |
| url | http://dx.doi.org/10.1155/2020/1816756 |
| work_keys_str_mv | AT paveltrojovsky onsomepropertiesofthehofstadtermertensfunction |