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Abstract

For any positive integer $k$ and any set $A$ of nonnegative integers, let $r_{1,k}(A,n)$ denote the number of solutions $(a_1,a_2)$ of the equation $n=a_1+ka_2$ with $a_1,a_2\in A$. Let $k,l\geq 2$ be two distinct integers. We prove that there exists a set $A\subseteq\mathbb N$ such that both $r_{1,k}(A,n)=r_{1,k}(\mathbb N\setminus A,n)$ and $r_{1,l}(A,n)=r_{1,l}(\mathbb N\setminus A,n)$ hold for all $n\geq n_0$ if and only if $\log k/\!\log l=a/b$ for some odd positive integers $a,b$, disproving a conjecture of Yang. We also show that for any set $A\subseteq\mathbb N$ satisfying $r_{1,k}(A,n)=r_{1,k}(\mathbb N\setminus A,n)$ for all $n\geq n_0$, we have $r_{1,k}(A,n)\rightarrow\infty$ as $n\to\infty$.