The aim of this article is to provoke discussion concerning arithmetic properties of the function $p_{d}(n)$ counting partitions of a positive integer n into dth powers, where $d\geq 2$ . Apart from results concerning the asymptotic behaviour of $p_{d}(n)$ , little is known. In the first part of the paper, we prove certain congruences involving functions counting various types of partitions into dth powers. The second part of the paper is experimental and contains questions and conjectures concerning the arithmetic behaviour of the sequence $(p_{d}(n))_{n\in \mathbb {N}}$ , based on computations of $p_{d}(n)$ for $n\leq 10^5$ for $d=2$ and $n\leq 10^{6}$ for $d=3, 4, 5$ .

中文翻译：

---

关于划分为 d 次幂的一些观察和推测

本文的目的是引发关于函数的算术性质的讨论$p_{d}(n)$计算正整数的分区n进入d次幂，其中$d\geq 2$. 除了关于渐近行为的结果$p_{d}(n)$，鲜为人知。在论文的第一部分，我们证明了涉及将各种类型的分区计数为的函数的某些同余。d权力。论文的第二部分是实验性的，包含有关序列算术行为的问题和猜想$(p_{d}(n))_{n\in \mathbb {N}}$，基于计算$p_{d}(n)$为了$n\leq 10^5$为了$d=2$和$n\leq 10^{6}$为了$d=3, 4, 5$.