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Abstract: Generalizing the concept of a perfect number, Sloane-s sequences of integersA083207 lists the sequence of integers $n$ with the property: the positivefactors of $n$ can be partitioned into two disjoint parts so that the sums ofthe two parts are equal. Following Clark et al., we shall call such integers,Zumkeller numbers. Generalizing this, Clark et al., call a number n ahalf-Zumkeller number if the positive proper factors of n can be partitionedinto two disjoint parts so that the sums of the two parts are equal.An extensive study of properties of Zumkeller numbers, half-Zumkeller numbersand their relation to practical numbers is undertaken in this paper.Clark et al., announced results about Zumkellers numbers and half-Zumkellernumbers and suggested two conjectures. In the present paper we shall settle oneof the conjectures, prove the second conjecture in some special cases and proveseveral results related to the second conjecture. We shall also show that ifthere is an even Zumkeller number that is not half-Zumkeller it should bebigger than 7233498900.

Author: K.P.S. Bhaskara Rao, Yuejian Peng

Source: https://arxiv.org/

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