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ON PARTITIONS OF NONNEGATIVE INTEGERS AND REPRESENTATION FUNCTIONS

Part of: Sequences and sets

Published online by Cambridge University Press:  11 December 2018

XIAO-HUI YAN Open the ORCID record for XIAO-HUI YAN [Opens in a new window]
XIAO-HUI YAN*
Affiliation:
School of Mathematical Sciences and Institute of Mathematics, Nanjing Normal University, Nanjing 210023, PR China email yanxiaohui_1992@163.com
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Abstract

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Let $\mathbb{N}$ be the set of all nonnegative integers. For any set $A\subset \mathbb{N}$, let $R(A,n)$ denote the number of representations of $n$ as $n=a+a^{\prime }$ with $a,a^{\prime }\in A$. There is no partition $\mathbb{N}=A\cup B$ such that $R(A,n)=R(B,n)$ for all sufficiently large integers $n$. We prove that a partition $\mathbb{N}=A\cup B$ satisfies $|R(A,n)-R(B,n)|\leq 1$ for all nonnegative integers $n$ if and only if, for each nonnegative integer $m$, exactly one of $2m+1$ and $2m$ is in $A$.

Keywords

representation functionpartitionSárközy problem

MSC classification

Primary: 11B34: Representation functions

Information

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 99 , Issue 3 , June 2019 , pp. 385 - 387
Copyright
© 2018 Australian Mathematical Publishing Association Inc. 

Footnotes

The author is supported by the National Natural Science Foundation of China, Grant No. 11771211.

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