Zumkeller数的一个推广：所有非负整数k的奇数k-IGMO数的无穷性
On a Generalization of Zumkeller Numbers: Infinitude of odd $k$-IGMO numbers for all non-negative integers $k$
AI总结：
本文推广Zumkeller数为k-IGMO数，证明所有非负整数k均存在无穷多个奇数k-IGMO数。
AI中文摘要：
本文受2025年国际伽马数学奥林匹克（IGMO）提出的问题启发，引入并研究了Zumkeller数的一种新推广，称为k-IGMO数。正整数n被定义为k-IGMO数，当且仅当它的正除数集合可划分为两个不相交的子集，且两子集元素的和相差k。根据该定义，经典Zumkeller数对应k=0的情况。本文的主要结果是：对于所有非负整数k，存在无穷多个奇数k-IGMO数。
英文摘要：
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 $k$.
