  Let $k\in\N_{\geq 2}$ and for given $m\in\Z\setminus\{0\}$ consider the
sequence $(S_{k,m}(n))_{n\in\N}$ defined by the power series expansion $$
\frac{1}{(1-x)^{m}}\prod_{i=0}^{\infty}\frac{1}{(1-x^{k^{i}})^{m}}=\sum_{n=0}^{\infty}S_{k,m}(n)x^{n}.
$$ The number $S_{k,m}(n)$ for $m\in\N_{+}$ has a natural combinatorial
interpretation: it counts the number of representations of $n$ as sums of
powers of $k$, where the part equal to $1$ takes one among $mk$ colors and each
part $>1$ takes $m(k-1)$ colors. We concentrate on the case when
$k=p\in\mathbb{P}$. Our main result is the computation of the exact value of
the $p$-adic valuation of $S_{p,m}(n)$. In particular, in each case the set of
values of $\nu_{p}(S_{p,m}(n))$ is finite and the maximum value is bounded by
$\op{max}\{\nu_{p}(m)+1,\nu_{p}(m+1)+1\}$. Our results can be seen as a
generalization of earlier work of Churchhouse and recent work of Gawron, Miska
and Ulas, and the present authors.
