On the differences between Zumkeller and $K$-layered numbers

TL;DR

This paper explores properties of Zumkeller and $k$-layered numbers, proving new theorems that extend the Green-Tao theorem to these classes of numbers, highlighting their structural differences and similarities.

Contribution

It introduces $k$-layered numbers as a generalization of Zumkeller numbers and proves stronger theorems related to their distribution.

Findings

Proves a theorem stronger than Green-Tao for Zumkeller numbers.

Establishes properties and differences of $k$-layered numbers.

Extends Green-Tao theorem to 4-layered numbers.

Abstract

A positive integer $n$ is said to be a Zumkeller number if the positive divisors of $n$ can be partitioned into two disjoint subsets of equal sum \cite{zumkeller}. In this paper, in the first section, we investigate differences between Zumkeller numbers and prove a theorem stronger than Green-Tao theorem for Zumkeller numbers. In the second section, we define $k$-layered numbers which are the generalization of Zumkeller numbers and investigate differences between $k$-layered numbers. We also prove a theorem stronger than Green-Tao theorem for 4-layered numbers.