PARTITIONS OF THE SET OF NONNEGATIVE INTEGERS WITH THE SAME REPRESENTATION FUNCTIONS

Abstract

Let$\mathbb{N}$be the set of all nonnegative integers. For a given set$S\subset \mathbb{N}$the representation function$R_{S}(n)$counts the number of solutions of the equation$n=s+s^{\prime }$with$s<s^{\prime }$and$s,s^{\prime }\in S$. We obtain some results on a problem of Chen and Lev [‘Integer sets with identical representation functions’,Integers16(2016), Article ID A36, 4 pages] about sets$A$and$B$such that$A\cup B=\mathbb{N}$,$A\cap B=r+m\mathbb{N}$and whose representation functions coincide.