Infinite Sets of Integers Whose Distinct Elements Do Not Sum to a Power

We first prove two results which both imply that for any sequence

of asymptotic density zero there exists an infinite sequence

such that the sum of any number of distinct elements of

does not belong to

Then, for any $\varepsilon >0,$ we construct an infinite sequence of positive integers $A=\{a_1<a_2<a_3<\dots\}$ satisfying $a_n < K(\varepsilon ) (1+\varepsilon )^n$ for each $n \in \mathbb{N}$ such that no sum of some distinct elements of

is a perfect square. Finally, given any finite set $U \subset \mathbb{N},$ we construct a sequence

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

with $u \in U,$ $v \in \mathbb{N}$ and $s \geq 2.$

Received November 13 2006; revised version received December 4 2006. Published in Journal of Integer Sequences December 4 2006.

Infinite Sets of Integers Whose Distinct Elements Do Not Sum to a Power

Artūras Dubickas and Paulius Šarka Department of Mathematics and Informatics Vilnius University Naugarduko 24 Vilnius LT-03225 Lithuania

Artūras Dubickas and Paulius Šarka
Department of Mathematics and Informatics
Vilnius University
Naugarduko 24
Vilnius LT-03225
Lithuania