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Representation functions with different weights

Volume 137 / 2014

Quan-Hui Yang Colloquium Mathematicum 137 (2014), 1-6 MSC: Primary 11B34; Secondary 05A17. DOI: 10.4064/cm137-1-1

Abstract

For any given positive integer $k$, and any set $A$ of nonnegative integers, let $r_{1, k}(A, n)$ denote the number of solutions of the equation $n=a_1+ka_2$ with $a_1, a_2\in A$. We prove that if $k,l$ are multiplicatively independent integers, i.e., $\log{k}/\log{l}$ is irrational, then there does not exist any set $A\subseteq \mathbb{N}$ such that both $r_{1,k}(A,n)=r_{1,k}(\mathbb{N}\setminus A,n)$ and $r_{1,l}(A,n)=r_{1,l}(\mathbb{N}\setminus A,n)$ hold for all $n\geq n_0$. We also pose a conjecture and two problems for further research.

Authors

  • Quan-Hui YangSchool of Mathematics and Statistics
    Nanjing University of Information Science and Technology
    Nanjing 210044, China
    e-mail

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