The Hofstadter consecutive-sum sequence omits infinitely many positive integers
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866910064837656576 |
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| author | Tang, Quanyu |
| author_facet | Tang, Quanyu |
| contents | Let $(a_n)_{n\ge 1}$ be the greedy self-generating sequence defined by $a_1=1$, $a_2=2$, and, for $k\ge 3$, by taking $a_k$ to be the least integer greater than $a_{k-1}$ that can be written as a sum of at least two consecutive earlier terms. Hofstadter asked about the asymptotic behavior of this sequence. In this paper we prove that $$ n+Ω(\log\log n)\le a_n \ll n^{4175/2506+o(1)}. $$ In particular, $(a_n)_{n\ge1}$ omits infinitely many positive integers, thereby settling a conjecture from the OEIS entry A005243. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2603_09939 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | The Hofstadter consecutive-sum sequence omits infinitely many positive integers Tang, Quanyu Number Theory Combinatorics Primary 11B83, Secondary 11D61 Let $(a_n)_{n\ge 1}$ be the greedy self-generating sequence defined by $a_1=1$, $a_2=2$, and, for $k\ge 3$, by taking $a_k$ to be the least integer greater than $a_{k-1}$ that can be written as a sum of at least two consecutive earlier terms. Hofstadter asked about the asymptotic behavior of this sequence. In this paper we prove that $$ n+Ω(\log\log n)\le a_n \ll n^{4175/2506+o(1)}. $$ In particular, $(a_n)_{n\ge1}$ omits infinitely many positive integers, thereby settling a conjecture from the OEIS entry A005243. |
| title | The Hofstadter consecutive-sum sequence omits infinitely many positive integers |
| topic | Number Theory Combinatorics Primary 11B83, Secondary 11D61 |
| url | https://arxiv.org/abs/2603.09939 |