For a set $A$, let $P(A)$ be the set of all finite subset sums of $A$. We prove that if a sequence $B=\{ 11\leq b_1<b_2<\cdots\}$ satisfies $b_2=3b_1+5$, $b_3=3b_2+2$ and $b_{ n+1}=3b_n+4b_{ n-1}$ for all $n\geq 3$, then there is a sequence of positive integers $A=\{ a_1<a_2<\cdots\}$ such that $P(A)=\mathbb{ N}\setminus B$. This result shows that the answer to the problem of Chen and Wu [`The inverse problem on subset sums', European. J. Combin. 34(2013), 841-845] is negative.
Categories: math.NT