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\def\eps{\varepsilon}
\def\N{\mathbb N}

\begin{abstract}
We first prove two results which both imply that for any sequence
$B$ of asymptotic density zero there exists an infinite sequence
$A$ such that the sum of any number of distinct elements of $A$
does not belong to $B.$ Then, for any $\eps>0,$ we construct an
infinite sequence of positive integers $A=\{a_1<a_2<a_3<\dots\}$
satisfying $a_n < K(\eps) (1+\eps)^n$ for each $n \in \N$ such
that no sum of some distinct elements of $A$ is a perfect square.
Finally, given any finite set $U \subset \N,$ we construct a
sequence $A$ of the same growth, namely, $a_n < K(\eps,U)
(1+\eps)^n$  for every $n \in \N$ such that no sum of its distinct
elements is equal to $uv^s$ with $u \in U,$ $v \in \N$ and $s \geq
2.$
\end{abstract}

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