On Zumkeller Numbers

TL;DR

This paper studies Zumkeller and half-Zumkeller numbers, proving one conjecture, partially addressing another, and establishing bounds, thereby advancing understanding of their properties and relationships to practical numbers.

Contribution

It proves one conjecture about Zumkeller numbers, addresses the second in special cases, and establishes bounds on even Zumkeller numbers not being half-Zumkeller.

Findings

Confirmed one conjecture about Zumkeller numbers.

Proved the second conjecture in specific cases.

Established that certain even Zumkeller numbers exceed 7,233,498,900.

Abstract

Generalizing the concept of a perfect number, Sloane's sequences of integers A083207 lists the sequence of integers $n$ with the property: the positive factors of $n$ can be partitioned into two disjoint parts so that the sums of the two parts are equal. Following Clark et al., we shall call such integers, Zumkeller numbers. Generalizing this, Clark et al., call a number n a half-Zumkeller number if the positive proper factors of n can be partitioned into two disjoint parts so that the sums of the two parts are equal. An extensive study of properties of Zumkeller numbers, half-Zumkeller numbers and their relation to practical numbers is undertaken in this paper. Clark et al., announced results about Zumkellers numbers and half-Zumkeller numbers and suggested two conjectures. In the present paper we shall settle one of the conjectures, prove the second conjecture in some special cases and prove several results related to the second conjecture. We shall also show that if there is an even Zumkeller number that is not half-Zumkeller it should be bigger than 7233498900.