Let N be the set of all nonnegative integers. For S ⊆ N and n ∈ N, let RS(n) denote the number of solutions of the equation n = s1 + s2, s1, s2 ∈ S and s1 < s2. Let A be the set of all nonnegative integers which contain an even number of digits 1 in their binary representations andThis solved a problem of Chen and Lev. In this paper, we prove that, if C ∪ D = [0, m] \ {r} with 0 < r < m, C ∩ D = ∅ and 0 ∈ C, then RC(n) = RD(n) for any nonnegative integer n if and only if there exists an integer l ≥ 2 such that m = 2 l , r = 2 l−

Let $\mathbb {N}$ be the set of all nonnegative integers. For $S\subseteq \mathbb {N}$ and $n\in \mathbb {N}$ , let $R_S(n)$ denote the number of solutions of the equation $n=s_1+s_2$ , $s_1,s_2\in S$ and $s_1<s_2$ . Let A be the set of all nonnegative integers which contain an even number of digits $1$ in their binary representations and $B=\mathbb {N}\setminus A$ . Put $A_l=A\cap [0,2^l-1]$ and $B_l=B\cap [0,2^l-1]$ . We prove that if $C \cup D=[0, m]\setminus \{r\}$ with $0<r<m$ , $C \cap D=\emptyset $ and $0 \in C$ , then $R_{C}(n)=R_{D}(n)$ for any nonnegative integer n if and only if there exists an integer $l \geq 1$ such that $m=2^{l}$ , $r=2^{l-1}$ , $C=A_{l-1} \cup (2^{l-1}+1+B_{l-1})$ and $D=B_{l-1} \cup (2^{l-1}+1+A_{l-1})$ . Kiss and Sándor [‘Partitions of the set of nonnegative integers with the same representation functions’, Discrete Math.340 (2017), 1154–1161] proved an analogous result when $C\cup D=[0,m]$ , $0\in C$ and $C\cap D=\{r\}$ .