On consecutive abundant numbers Article Swipe

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YOU? · · 2016 · Open Access · · DOI: https://doi.org/10.48550/arxiv.1603.06176

A positive integer $n$ is called an abundant number if $σ(n)\ge 2n$, where $σ(n)$ is the sum of all positive divisors of $n$. Let $E(x)$ be the largest number of consecutive abundant numbers not exceeding $x$. In 1935, P. Erd\H os proved that there are two positive constants $c_1$ and $c_2$ such that $c_1\log\log\log x\le E(x)\le c_2\log\log\log x$. In this paper, we resolve this old problem by proving that, $E(x)/\log \log\log x$ tends to a limit as $x\to +\infty$, and the limit value has an explicit form which is between $3$ and $4$.

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Combinatorics Integer (computer science) Limit (mathematics) Mathematics Value (mathematics) Binary logarithm Sigma Log-log plot Physics Statistics Mathematical analysis Computer science Quantum mechanics Programming language

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Type: preprint
Language: en
Landing Page: http://arxiv.org/abs/1603.06176
PDF: https://arxiv.org/pdf/1603.06176
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OpenAlex ID: https://openalex.org/W2310337644

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DOI: https://doi.org/10.48550/arxiv.1603.06176

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Title: On consecutive abundant numbers

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Publication year: 2016

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Publication date: 2016-03-20

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Authors: Yong-Gao Chen, Hui Lv

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Landing page: https://arxiv.org/abs/1603.06176

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Concepts: Combinatorics, Integer (computer science), Limit (mathematics), Mathematics, Value (mathematics), Binary logarithm, Sigma, Log-log plot, Physics, Statistics, Mathematical analysis, Computer science, Quantum mechanics, Programming language

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