December 9, 2014
Using an extension of the abundancy index to imaginary quadratic rings with unique factorization, we define what we call $n$-powerfully perfect numbers in these rings. This definition serves to extend the concept of perfect numbers that have been defined and studied in the integers. We investigate the properties of $2$-powerfully perfect numbers in the rings $\mathcal O_{\mathbb{Q}(\sqrt{-1})}$, $\mathcal O_{\mathbb{Q}(\sqrt{-2})}$, and $\mathcal O_{\mathbb{Q}(\sqrt{-7})}$, the three imaginary quadratic rings with unique factorization in which $2$ is not a prime.
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June 17, 2015
Using an extension of the abundancy index to imaginary quadratic rings that are unique factorization domains, we investigate what we call $n$-powerfully $t$-perfect numbers in these rings. This definition serves to extend the concept of multiperfect numbers that have been defined and studied in the integers. At the end of the paper, as well as at various points throughout the paper, we point to some potential areas for further research.
June 17, 2015
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 form, we investigate what we call $n$-powerfully solitary numbers in these rings. This definition serves to extend the concept of solitary numbers, which have been defined and studied in the integers. We end with some open questions and a con...
December 9, 2014
A unitary divisor $c$ of a positive integer $n$ is a positive divisor of $n$ that is relatively prime to $\displaystyle{\frac{n}{c}}$. For any integer $k$, the function $\sigma_k^*$ is a multiplicative arithmetic function defined so that $\sigma_k^*(n)$ is the sum of the $k^{th}$ powers of the unitary divisors of $n$. We provide analogues of the functions $\sigma_k^*$ in imaginary quadratic rings that are unique factorization domains. We then explore properties of what we cal...
April 24, 2020
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 the ring of Gaussian integers ($\mathbb{Z}[i]$). We first show that this generalization possesses many of the useful properties of the traditional abundancy index in $\mathbb{Z}$. We then investigate $k$-powerful $\tau$-perfect numbers and...
August 5, 2021
In this note, we continue an approach pursued in an earlier paper of the second author and thereby attempt to produce an improved lower bound for the sum $I(q^k) + I(n^2)$, where $q^k n^2$ is an odd perfect number with special prime $q$ and $I(x)$ is the abundancy index of the positive integer $x$. In particular, this yields an upper bound for $k$.
September 29, 2023
Let $N$ be an odd perfect number. Then, Euler proved that there exist some integers $n, \alpha$ and a prime $q$ such that $N = n^{2}q^{\alpha}$, $q \nmid n$, and $q \equiv \alpha \equiv 1 \bmod 4$. In this note, we prove that the ratio $\frac{\sigma(n^{2})}{q^{\alpha}}$ is neither a square nor a square times a single prime unless $\alpha = 1$. It is a direct consequence of a certain property of the Diophantine equation $2ln^{2} = 1+q+ \cdots +q^{\alpha}$, where $l$ denotes on...
August 1, 2010
In this paper some new ways of generalizing perfect numbers are investigated, numerical results are presented and some conjectures are established.
February 29, 2016
One of the many number theoretic topics investigated by the ancient Greeks was perfect numbers, which are positive integers equal to the sum of their proper positive integral divisors. Mathematicians from Euclid to Euler investigated these mysterious numbers. We present results on perfect numbers in the ring of Eisenstein integers.
June 2, 2021
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 superab...
September 28, 2016
We will show the two following results: If there existe an odd perfect number $n$ of prime decomposition $n=p_1^{\alpha_1} \ldots p_k^{\alpha_k}q^\beta$, where the $\alpha_i$ are even, the $\beta$ are odd and $q \equiv 5 \mod 8$. Then there is at least one $p_i$, $1 \leq i \leq k$ that is not a square in $\mathbb{Z}/q\mathbb{Z}$. More precisely there is an odd number of $p_i$ that are not squares in $\mathbb{Z}/q\mathbb{Z}$. If there exist an odd perfect number $n$ of prime d...