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arXiv · 2608.13890

On a Generalization of Zumkeller Numbers: Infinitude of odd $k$-IGMO numbers for all non-negative integers $k$

Ting Hon Stanford Li
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Abstract

In this paper, we introduce and investigate a novel generalization of Zumkeller numbers termed $k$-IGMO numbers, inspired by a problem proposed in the International Gamma Mathematical Olympiad (IGMO) 2025. A positive integer n is defined as a $k$-IGMO number if its set of positive divisors can be partitioned into two disjoint subsets whose elements have sums differing by $k$. Under this definition, classical Zumkeller numbers correspond to the case where $k = 0$. The main result of the paper is the existence of infinitely many odd $k$-IGMO numbers for all non-negative integers $k$.

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Ting Hon Stanford Li. 2026-08-14. On a Generalization of Zumkeller Numbers: Infinitude of odd $k$-IGMO numbers for all non-negative integers $k$. https://arxiv.org/abs/2608.13890

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