关于Zumkeller数
On Zumkeller Numbers
摘要：
Sloane的整数序列A083207列出了整数序列$n$的性质：$n$的正因子可以划分为两个不相交的部分，使得两个部分的和相等。继克拉克等人之后，我们称之为Zumkeller数。概括这一点，克拉克等人，称一个数n为半Zumkeller数，如果n的正真因子可以被划分为两个不相交的部分，使得这两个部分的和相等。本文对Zumkeller数、半Zumkeller数的性质及其与实际数的关系进行了广泛的研究。Clark等人，公布了关于Zumkeller数和半Zumkeller数的结果，并提出了两个猜想。本文将解决其中的一个猜想，在某些特殊情况下证明第二个猜想，并证明与第二个猜想有关的几个结果。我们还将证明，如果有一个不是半Zumkeller的偶数Zumkeller数，它应该大于7233498900。
英文摘要：
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.
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