Geodesic
Infinite sets of integers whose distinct elements do not sum to a power
Journal of integer sequences, Tome 9 (2006) no. 4
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 $\varepsilon >0,$ we construct an infinite sequence of positive integers $A=\{a_1$ satisfying $a_n K(\varepsilon ) (1+\varepsilon )^n$ for each $n \in \mathbb{N}$ such that no sum of some distinct elements of $A$ is a perfect square. Finally, given any finite set $U \subset \mathbb{N},$ we construct a sequence $A$ of the same growth, namely, $a_n K(\varepsilon ,U) (1+\varepsilon )^n$ for every $n \in \mathbb{N}$ such that no sum of its distinct elements is equal to $uv^s$ with $u \in U,v \in \mathbb{N}$ and $s \geq 2.$
Classification : 11A99, 11B05, 11B99
Keywords: infinite sequence, perfect square, power, asymptotic density, sumset 
@article{JIS_2006__9_4_a0,
     author = {Dubickas,  Art\={u}ras and \v{S}arka,  Paulius},
     title = {Infinite sets of integers whose distinct elements do not sum to a power},
     journal = {Journal of integer sequences},
     year = {2006},
     volume = {9},
     number = {4},
     zbl = {1119.11011},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/JIS_2006__9_4_a0/}
}
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Dubickas,  Artūras; Šarka,  Paulius. Infinite sets of integers whose distinct elements do not sum to a power. Journal of integer sequences, Tome 9 (2006) no. 4. http://geodesic.mathdoc.fr/item/JIS_2006__9_4_a0/