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Partitions of natural numbers with the same weighted representation functions

Volume 159 / 2020

Ya-Li Li, Wu-Xia Ma 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.

Authors

  • Ya-Li LiSchool of Mathematics and Statistics
    Henan University
    Kaifeng 475001, P.R. China
    e-mail
  • Wu-Xia MaSchool of Mathematical Sciences
    and Institute of Mathematics
    Nanjing Normal University
    Nanjing 210023, P.R. China
    e-mail

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