Unitary Multiperfect Numbers in Certain Quadratic Rings

Abstract

A unitary divisor $c$ of a positive integer $n$ is a positive divisor of $n$ that is relatively prime to $\frac{n}{c}$. For any integer $k$, the function $\ddot{I}{ƒ}_k^{\ast }$ is a multiplicative arithmetic function defined so that $\ddot{I}{ƒ}_k^{\ast }(n)$ is the sum of the $k^{th}$ powers of the unitary divisors of $n$. We provide analogues of the functions $\ddot{I}{ƒ}_k^{\ast }$ 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.

14 pages, 0 figures, Supported by National Science Foundation grant no. 1262930. arXiv admin note: text overlap with arXiv:1412.3072