  Let $k,\ell\geq2$ be fixed integers and $C$ be an effectively computable
constant depending only on $k$ and $\ell$. In this paper, we prove that all
solutions of the equation $(x+1)^{k}+(x+2)^{k}+...+(\ell x)^{k}=y^{n}$ in
integers $x,y,n$ with $x,y\geq1, n\geq2, k\neq3$ and $\ell\equiv 1 \pmod 2$
satisfy $\max\{x,y,n\}<C$. The case when $\ell$ is even has already been
completed by Soydan (Publ. Math. Debrecen 91 (2017), pp. 369-382).
