Partitions of natural numbers with the same weighted representation functions
Volume 159 / 2020
Colloquium Mathematicum 159 (2020), 1-5
MSC: Primary 11B34; Secondary 05A17.
DOI: 10.4064/cm7605-11-2018
Published online: 6 September 2019
Abstract
For a set $A$ of nonnegative integers and two positive integers $k_1, k_2$, let $r_{k_1,k_2}(A,n)$ be the number of solutions of $n=k_1a_1+k_2a_2$, where $a_1, a_2\in A$. In 2012, Yang and Chen determined all pairs $k_1$ and $k_2$ for which there exists a set $A$ of nonnegative integers such that $r_{k_1,k_2}(A, n)=r_{k_1,k_2}(\mathbb {N}\setminus A,n)$ for all sufficiently large integers $n$. We use generating functions to give new proofs of the results by Yang and Chen.
