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Representation functions with different weights

Quan-Hui Yang (2014)

Colloquium Mathematicae

For any given positive integer k, and any set A of nonnegative integers, let r 1 , k ( A , n ) denote the number of solutions of the equation n = a₁ + ka₂ with a₁,a₂ ∈ A. We prove that if k,l are multiplicatively independent integers, i.e., log k/log l is irrational, then there does not exist any set A ⊆ ℕ such that both r 1 , k ( A , n ) = r 1 , k ( A , n ) and r 1 , l ( A , n ) = r 1 , l ( A , n ) hold for all n ≥ n₀. We also pose a conjecture and two problems for further research.

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