  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 $d\geq 2$. 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 $(p_{d}(n))_{n\in\N}$. They based on our computations of
$p_{d}(n)$ for $n\leq 10^5$ in case of $d=2$, and $n\leq 10^{6}$ for $d=3, 4,
5$.
