PARTITIONS OF NATURAL NUMBERS AND THEIR WEIGHTED REPRESENTATION FUNCTIONS

Abstract

Abstract 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 g be a fixed integer. We prove that if $k_1$ and $k_2$ are two integers with $2\le k_1<k_2$ and $(k_1,k_2)=1$ , then there does not exist any set $A\subseteq \mathbb {N}$ such that $R_{k_1,k_2}(A,n)-R_{k_1,k_2}(\mathbb {N}\setminus A,n)=g$ for all sufficiently large integers n, and if $1=k_1<k_2$ , then there exists a set A such that $R_{k_1,k_2}(A,n)-R_{k_1,k_2}(\mathbb {N}\setminus A,n)=1$ for all positive integers n.