Unitary Multiperfect Numbers in Certain Quadratic Rings

TL;DR

This paper extends the concept of unitary divisor functions to imaginary quadratic rings that are unique factorization domains, exploring properties of $n$-powerfully unitarily $t$-perfect numbers and suggesting future research directions.

Contribution

It introduces analogues of the $\sigma_k^*$ functions in quadratic rings and studies the properties of unitarily perfect numbers within this new setting.

Findings

Defined $\sigma_k^*$ analogues in quadratic rings

Explored properties of $n$-powerfully unitarily $t$-perfect numbers

Provided avenues for further research

Abstract

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 call $n$-powerfully unitarily $t$-perfect numbers, analogues of the unitary multiperfect numbers that have been defined and studied in the integers. We end with a list of several opportunities for further research.