Measuring Abundance with Abundancy Index

A positive integer \(n\) is called perfect if \( \sigma(n)=2n\), where \(\sigma(n)\) denote the sum of divisors of \(n\). In this paper we study the ratio \(\frac{\sigma(n)}{n}\). We define the function Abundancy Index \(I:\mathbb{N} \to \mathbb{Q}\) with \(I(n)=\frac{\sigma(n)}{n}\). Then we study...

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Published in:arXiv.org 2021-08
Main Authors: Guha, Kalpok, Ghosh, Sourangshu
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description A positive integer \(n\) is called perfect if \( \sigma(n)=2n\), where \(\sigma(n)\) denote the sum of divisors of \(n\). In this paper we study the ratio \(\frac{\sigma(n)}{n}\). We define the function Abundancy Index \(I:\mathbb{N} \to \mathbb{Q}\) with \(I(n)=\frac{\sigma(n)}{n}\). Then we study different properties of the Abundancy Index and discuss the set of Abundancy Index. Using this function we define a new class of numbers known as superabundant numbers. Finally, we study superabundant numbers and their connection with Riemann Hypothesis.
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title Measuring Abundance with Abundancy Index
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