Journal of Integer Sequences, Vol. 9 (2006), Article 06.4.8 |

Department of Mathematics and Informatics

Vilnius University

Naugarduko 24

Vilnius LT-03225

Lithuania

**Abstract:**

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
we construct an
infinite sequence of positive integers
satisfying
for each
such
that no sum of some distinct elements of is a perfect square.
Finally, given any finite set
we construct a
sequence of the same growth, namely,
for every
such that no sum of its distinct
elements is equal to with
and

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

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