# 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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topZhenhua 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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