$\tau$-Norm-Perfect and $\tau$-Perfect Eisenstein Integers for $\tau =\omega +2$ and $2$

Abstract

Using Robert Spira's \cite{D} definitions of complex Mersenne numbers and the complex sum-of-divisors function, we characterize $(\omega +2)$-norm-perfect and $(\omega +2)$-perfect numbers that are divisble by $\omega +2$ and prove the nonexistence of $2$-norm-perfect numbers that are divisible by $2$ in the Eisenstein integers.