# A note on representation functions with different weights

Colloquium Mathematicae (2016)

• Volume: 143, Issue: 1, page 105-112
• ISSN: 0010-1354

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## Abstract

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For any positive integer k and any set A of nonnegative integers, let ${r}_{1,k}\left(A,n\right)$ denote the number of solutions (a₁,a₂) of the equation n = a₁ + ka₂ with a₁,a₂ ∈ A. Let k,l ≥ 2 be two distinct integers. We prove that there exists a set A ⊆ ℕ such that both ${r}_{1,k}\left(A,n\right)={r}_{1,k}\left(ℕ\setminus A,n\right)$ and ${r}_{1,l}\left(A,n\right)={r}_{1,l}\left(ℕ\setminus A,n\right)$ hold for all n ≥ n₀ if and only if log k/log l = a/b for some odd positive integers a,b, disproving a conjecture of Yang. We also show that for any set A ⊆ ℕ satisfying ${r}_{1,k}\left(A,n\right)={r}_{1,k}\left(ℕ\setminus A,n\right)$ for all n ≥ n₀, we have ${r}_{1,k}\left(A,n\right)\to \infty$ as n → ∞.

## How to cite

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Zhenhua Qu. "A note on representation functions with different weights." Colloquium Mathematicae 143.1 (2016): 105-112. <http://eudml.org/doc/283601>.

@article{ZhenhuaQu2016,
abstract = {For any positive integer k and any set A of nonnegative integers, let $r_\{1,k\}(A,n)$ denote the number of solutions (a₁,a₂) of the equation n = a₁ + ka₂ with a₁,a₂ ∈ A. Let k,l ≥ 2 be two distinct integers. We prove that there exists a set A ⊆ ℕ such that both $r_\{1,k\}(A,n) = r_\{1,k\}(ℕ ∖ A,n)$ and $r_\{1,l\}(A,n) = r_\{1,l\}(ℕ ∖ A,n)$ hold for all n ≥ n₀ if and only if log k/log l = a/b for some odd positive integers a,b, disproving a conjecture of Yang. We also show that for any set A ⊆ ℕ satisfying $r_\{1,k\}(A,n) = r_\{1,k\}(ℕ ∖ A,n)$ for all n ≥ n₀, we have $r_\{1,k\}(A,n) → ∞$ as n → ∞.},
author = {Zhenhua Qu},
journal = {Colloquium Mathematicae},
keywords = {representation function; partition; Sárközy problem},
language = {eng},
number = {1},
pages = {105-112},
title = {A note on representation functions with different weights},
url = {http://eudml.org/doc/283601},
volume = {143},
year = {2016},
}

TY - JOUR
AU - Zhenhua Qu
TI - A note on representation functions with different weights
JO - Colloquium Mathematicae
PY - 2016
VL - 143
IS - 1
SP - 105
EP - 112
AB - For any positive integer k and any set A of nonnegative integers, let $r_{1,k}(A,n)$ denote the number of solutions (a₁,a₂) of the equation n = a₁ + ka₂ with a₁,a₂ ∈ A. Let k,l ≥ 2 be two distinct integers. We prove that there exists a set A ⊆ ℕ such that both $r_{1,k}(A,n) = r_{1,k}(ℕ ∖ A,n)$ and $r_{1,l}(A,n) = r_{1,l}(ℕ ∖ A,n)$ hold for all n ≥ n₀ if and only if log k/log l = a/b for some odd positive integers a,b, disproving a conjecture of Yang. We also show that for any set A ⊆ ℕ satisfying $r_{1,k}(A,n) = r_{1,k}(ℕ ∖ A,n)$ for all n ≥ n₀, we have $r_{1,k}(A,n) → ∞$ as n → ∞.
LA - eng
KW - representation function; partition; Sárközy problem
UR - http://eudml.org/doc/283601
ER -

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