On the differences between Zumkeller and $K$-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.