Some Results on Generalized Multiplicative Perfect Numbers

TL;DR

This paper introduces new classes of generalized multiplicative perfect numbers, explores their properties, and characterizes specific types, extending previous work by Sándor and others in number theory.

Contribution

It defines $k$-multiplicatively $e$-perfect and $e$-superperfect numbers and characterizes $k$-$T_0T^*$-perfect numbers, advancing the understanding of generalized perfect numbers.

Findings

Defined new classes of generalized perfect numbers.

Proved properties of $k$-multiplicatively $e$-perfect and superperfect numbers.

Characterized $k$-$T_0T^*$-perfect numbers in detail.

Abstract

In this article, based on ideas and results by J. S\'andor (2001, 2004), we define $k$-multiplicatively $e$-perfect numbers and $k$-multiplicatively $e$-superperfect numbers and prove some results on them. We also characterize the $k$-$T_0T^\ast$-perfect numbers defined by Das and Saikia (2013) in details.