Some observations and speculations on partitions into $d$-th powers

Abstract

The aim of this note is to provoke discussion concerning arithmetic properties of function $p_d(n)$ counting partitions of an positive integer $n$ into $d$-th powers, where $d\ge 2$. Besides results concerning the asymptotic behavior of $p_d(n)$ a little is known. In the first part of the paper, we prove certain congruences involving functions counting various types of partitions into $d$-th powers. The second part of the paper has experimental nature and contains questions and conjectures concerning arithmetic behavior of the sequence $(p_d(n){)}_{n\in \mathbb{N}}$. They based on our computations of $p_d(n)$ for $n\le 10^5$ in case of $d=2$, and $n\le 10^6$ for $d=3,4,5$.

8 pages, revised version will appear in Bull. Aust. Math. Society