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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2025
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2506.04883 |
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| _version_ | 1866910010286538752 |
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| author | Kovač, Vjekoslav Luca, Florian |
| author_facet | Kovač, Vjekoslav Luca, Florian |
| contents | Denote $f(n):=\sum_{1\le k\le n} τ(2^k-1)$, where $τ$ is the number of divisors function. Motivated by a question of Paul Erdős, we show that the sequence of ratios $f(2n)/f(n)$ is unbounded. We also present conditional results on the divergence of this sequence to infinity. Finally, we test numerically both the conjecture $f(2n)/f(n)\to\infty$ and our sufficient conditions for it to hold. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2506_04883 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On the number of divisors of Mersenne numbers Kovač, Vjekoslav Luca, Florian Number Theory Denote $f(n):=\sum_{1\le k\le n} τ(2^k-1)$, where $τ$ is the number of divisors function. Motivated by a question of Paul Erdős, we show that the sequence of ratios $f(2n)/f(n)$ is unbounded. We also present conditional results on the divergence of this sequence to infinity. Finally, we test numerically both the conjecture $f(2n)/f(n)\to\infty$ and our sufficient conditions for it to hold. |
| title | On the number of divisors of Mersenne numbers |
| topic | Number Theory |
| url | https://arxiv.org/abs/2506.04883 |