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\begin{abstract}
Given integers $k\ge 2$ and $\ell\ge 3$, let $S_{k,\ell}^*$ stand for
the set of those positive integers $n$ which can be written as
$n=p_1^k+p_2^k+\ldots+p_\ell^k$, where $p_1,p_2,\ldots,p_\ell$ are
distinct prime factors of $n$. We study the properties of the sets
$S^*_{k,\ell}$ and we show in particular that, given any odd $\ell\ge
3$, $\displaystyle{\#\bigcup_{k=2}^\infty S_{k,\ell}^*=+\infty}$.
\end{abstract}

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