| For any positive integers $k_1,k_2$ and any set $A\subseteq \mathbb{N}$, let $R_{k_1,k_2}(A,n)$ be the number of solutions of the equation $n=k_1a_1+k_2a_2$ with $a_1,a_2\in A$. Let $\bar{A}=\mathbb{N}\backslash A$. Yang and Chen proved that if $k_1$ and $k_2$ are two integers with $k_2>k_1\geq 2$ and $(k_1,k_2)=1$, then there does not exist any set $A\subseteq\mathbb{N}$ such that $R_{k_1,k_2}(A,n)=R_{k_1,k_2}(\bar{A},n)$ for all sufficiently large integers $n$. For two integers $k>1$ and $t\geq1$, define $f_{k}(t)$ to be the number of sets $A\subseteq\mathbb{N}$ such that $R_{1,k}(A,n)=R_{1,k}(\bar{A},n)$ holds for all integers $n\geq t$. Yang and Chen proved that $f_{k}(t)$ is finite and $\lim_{t\rightarrow\infty}\frac{\log f_{k}(t)}{t}=\log2$. They also asked if it is true that for any integers $k,l>1$ there exists $t_0(k,l)$ such that $f_k(t)=f_l(t)$ for all integers $t\geq t_0$. In this paper, we give the exact formula of $f_{k}(t)$ when $t\leq k$, which implies that $f_k(t)=f_l(t)$ for all integers $t\leq \min\{k,l\}$. |
