# 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 → ∞.

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