Unitary Multiperfect Numbers in Certain Quadratic Rings

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A unitary divisor c of a positive integer n is a positive divisor of n that is relatively prime to n/c. For any integer k, the function σₖ* is a multiplicative arithmetic function defined so that σₖ*(n) is the sum of the kth powers of the unitary divisors of n. We provide analogues of the functions σₖ* 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.