ON A PROBLEM OF CHEN AND LEV
ON A PROBLEM OF CHEN AND LEV
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DOI：
10.1017/s0004972718001107
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发表时间：
2018-11
影响因子：
0.7
通讯作者：
Shi-Qiang Chen;M. Tang;Quan-Hui Yang
中科院分区：
文献类型：
--
作者：
Shi-Qiang Chen;M. Tang;Quan-Hui Yang
For a given set $S\subset \mathbb{N}$ , $R_{S}(n)$ is the number of solutions of the equation $n=s+s^{\prime },sr\geq 0$ and that $A$ and $B$ are sets with $A\cup B=\mathbb{N}$ and $A\cap B=\{r+mk:k\in \mathbb{N}\}$ . We prove that if $R_{A}(n)=R_{B}(n)$ for all positive integers $n$ , then there exists an integer $l\geq 1$ such that $r=2^{2l}-1$ and $m=2^{2l+1}-1$ . This solves a problem of Chen and Lev [‘Integer sets with identical representation functions’, Integers 16 (2016), A36] under the condition $m>r$ .
For a given set $S\subset \mathbb{N}$ , $R_{S}(n)$ is the number of solutions of the equation $n=s+s^{\prime },sr\geq 0$ and that $A$ and $B$ are sets with $A\cup B=\mathbb{N}$ and $A\cap B=\{r+mk:k\in \mathbb{N}\}$ . We prove that if $R_{A}(n)=R_{B}(n)$ for all positive integers $n$ , then there exists an integer $l\geq 1$ such that $r=2^{2l}-1$ and $m=2^{2l+1}-1$ . This solves a problem of Chen and Lev [‘Integer sets with identical representation functions’, Integers 16 (2016), A36] under the condition $m>r$ .