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The estimate on the natural density of integers $n$ for which $σ(kn+r_1) geq σ(kn+r_2)$ | | Xin-qi Luo
; Chen-kai Ren
; | | Date: |
1 Nov 2023 | | Abstract: | For any positive integer $n$, let $sigma(n)=sum_{dmid n} d$. In 1936, P. Erdös proved that the natural density of the set of positive integers $n$ for which $sigma(n+1) geq sigma(n)$ is $frac{1}{2}$. In 2020, M. Kobayashi and T. Trudgian showed that the natural density of positive integers $n$ with $sigma(2n+1) geq sigma(2n)$ is between 0.053 and 0.055. In this paper, we generalize M. Kobayashi and T. Trudgian's result cite{KT20} and give a new estimate on the natural density of positive integers $n$ for which $sigma(kn+r_1) geq sigma(kn+r_2)$. We also calculate some special cases with certain $k,r_1$ and $r_2$. | | Source: | arXiv, 2311.00295 | | Services: | Forum | Review | PDF | Favorites | |
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