On Zumkeller numbers

Abstract: Text: A positive integer n is perfect if the sum of the proper positive divisors of n equals n. Generalizing this we call n a Zumkeller number if the set of its positive divisors can be partitioned into two disjoint subsets of equal sum. Similarly we call n a half-Zumkeller number if the set of its proper positive divisors can be so partitioned. A study of Zumkeller numbers, half-Zumkeller numbers and their relation to practical numbers is undertaken in this paper. Clark et al. (2008) [1] announced some results about Zumkeller 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 is bigger than . Video: For a video summary of this paper, please click here or visit http://www.youtube.com/watch?v=z85qyvIorBE. [Copyright &y& Elsevier]