Related papers: Analysis of the Ratio $D(n)/n$

We show that $n$ is almost perfect if and only if $I(n) - 1 < D(n) \leq I(n)$, where $I(n)$ is the abundancy index of $n$ and $D(n)$ is the deficiency of $n$. This criterion is then extended to the case of integers $m$ satisfying $D(m)>1$.

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 $I:\mathbb{N} \to \mathbb{Q}$…

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…

Abundancy index refers to the ratio of the sum of the divisors of a number to the number itself. It is a concept of great importance in defining friendly and perfect numbers. Here, we describe a suitable generalization of abundancy index to…

We begin by introducing an extension of the traditional abundancy index to imaginary quadratic rings with unique factorization. After showing that many of the properties of the traditional abundancy index continue to hold in our extended…

It is largely taken for granted that differential abundance analysis is, by default, the best first step when analyzing genomic data. We argue that this is not necessarily the case. In this article, we identify key limitations that are…

In recent years, the concepts of ``diversity'' and ``inclusion'' have attracted considerable attention across a range of fields, encompassing both social and biological disciplines. To fully understand these concepts, it is critical to not…

Pairwise comparisons between alternatives are a well-established tool to decompose decision problems into smaller and more easily tractable sub-problems. However, due to our limited rationality, the subjective preferences expressed by…

The aim of this note is to provoke discussion concerning arithmetic properties of function $p_{d}(n)$ counting partitions of an positive integer $n$ into $d$-th powers, where $d\geq 2$. Besides results concerning the asymptotic behavior of…

In this note, we revisit a classical problem related to the density of nonlinear statistics. We obtain a new representation of densities and, for the first time, a necessary and sufficient condition for the existence of densities is…

Let D = (D_n)_{n\ge1} be an elliptic divisibility sequence. We study the set S(D) of indices n satisfying n | D_n. In particular, given an index n in S(D), we explain how to construct elements nd in S(D), where d is either a prime divisor…

Data redundancy is ubiquitous in the inputs and intermediate results of Deep Neural Networks (DNN). It offers many significant opportunities for improving DNN performance and efficiency and has been explored in a large body of work. These…

The Gini index is a function that attempts to measure the amount of inequality in the distribution of a finite resource throughout a population. It is commonly used in economics as a measure of inequality of income or wealth. We define a…

The estimation of the ratio of two density probability functions is of great interest in many statistics fields, including causal inference. In this study, we develop an ensemble estimator of density ratios with a novel loss function based…

This paper reviews the modularity index and suggests an alternative index of the quality of a division of a network into subsets.

A fundamental challenge in quantum resource theory is to establish operational interpretations by quantifying the advantage that quantum resources provide in specific tasks. Conventional resource theories, however, have inherent limitations…

We survey the different properties of an intuitive notion of redundancy, as a function of the precise semantics given to the notion of partial implication. The final version of this survey will appear in the Proceedings of the Int. Conf.…

We examine the convergence properties of sequences of nonnegative real numbers that satisfy a particular class of recursive inequalities, from the perspective of proof theory and computability theory. We first establish a number of results…

We consider a bivariate polynomial that generalizes both the length and reflection length generating functions in a finite Coxeter group. In seeking a combinatorial description of the coefficients, we are led to the study of a new Mahonian…

Let $\{U(m)\}_{m\in \N}$ and $\{V(n)\}_{n\in \N}$ be linear recurrence sequences. It is a well-known Diophantine problem to determine the finiteness of the set of natural numbers $n$ such that the ratio $U(n)/V(n)$ is an integer. We study…