2017 CMO 第 4 题

Let $D_n$ be the set of divisors of $n$ . Find all natural $n$ such that it is possible to split $D_n$ into two disjoint sets $A$ and $G$ , both containing at least three elements each, such that the elements in $A$ form an arithmetic progression while the elements in $G$ form a geometric progression.

令 $D_n$ 为 $n$ 的除数集合。找到所有自然的$n$，使得可以将$D_n$分成两个不相交的集合$A$和$G$，每个集合都至少包含三个元素，使得$A$中的元素形成算术级数，而$G$中的元素形成几何级数。