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arXiv · 1803.10816

Generalizing the Abundancy of an Integer

David C. Luo
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Abstract

The abundancy index of a positive integer is the ratio between the sum of its divisors and itself. We generalize previous results on abundancy indices by defining a two-variable abundancy index function as $I(x,n)\colon\mathbb{Z^+}\times\mathbb{Z^+}\to\mathbb{Q}$ where $I(x,n)=\frac{\sigma_x(n)}{n^x}$. Specifically, we extend limiting properties of the abundancy index and construct sufficient conditions for rationals greater than one that fail to be in the image of the function $I(x,n)$.

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David C. Luo. 2018-03-28. Generalizing the Abundancy of an Integer. https://arxiv.org/abs/1803.10816

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