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PARTITIONS OF NATURAL NUMBERS AND THEIR WEIGHTED REPRESENTATION FUNCTIONS
- Part of
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- Journal:
- Bulletin of the Australian Mathematical Society / Volume 110 / Issue 1 / August 2024
- Published online by Cambridge University Press:
- 27 October 2023, pp. 12-18
- Print publication:
- August 2024
-
- Article
- Export citation
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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.
ON PARTITIONS OF NONNEGATIVE INTEGERS AND REPRESENTATION FUNCTIONS
- Part of
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- Journal:
- Bulletin of the Australian Mathematical Society / Volume 99 / Issue 3 / June 2019
- Published online by Cambridge University Press:
- 11 December 2018, pp. 385-387
- Print publication:
- June 2019
-
- Article
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- You have access
- Export citation
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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 
$A$.
