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PARTITIONS OF NATURAL NUMBERS AND THEIR WEIGHTED REPRESENTATION FUNCTIONS

Published online by Cambridge University Press:  27 October 2023

SHUANG-SHUANG LI
Affiliation:
Office of Scientific Research, Anhui Normal University, Wuhu 241002, PR China e-mail: ddlshuang@163.com
YU-QING SHAN
Affiliation:
School of Mathematics and Statistics, Anhui Normal University, Wuhu 241002, PR China e-mail: syq357660@163.com
XIAO-HUI YAN*
Affiliation:
School of Mathematics and Statistics, Anhui Normal University, Wuhu 241002, PR China

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.

MSC classification

Type
Research Article
Copyright
© The Author(s), 2023. Published by Cambridge University Press on behalf of Australian Mathematical Publishing Association Inc.

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Footnotes

This work is supported by the National Natural Science Foundation of China (Grant Nos. 12101009 and 12371005), Anhui Provincial Natural Science Foundation (Grant No. 2108085QA02) and University Natural Science Research Project of Anhui Province (Grant No. 2022AH050171).

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