Partitions of natural numbers with the same representation functions (Q841249)

Let \(\mathbb{N}\) be the set of nonnegative integers. For a set \(A \subseteq \mathbb{N}\) let \(R_{2}(A, n)\) and \(R_{3}(A, n)\) be the number of representations of \(n\) as the sum of two terms \(a\), \(b\) from the set \(A\) with \(a < b\), \(a \leq b\) respectively. The authors prove the following theorems about the above functions. If \(R_{3}(A, n) = R_{3}(\mathbb{N} \setminus A, n)\) for all large enough \(n\), then \(R_{3}(A, n) = R_{3}(\mathbb{N} \setminus A, n) \geq 1\) for all large enough \(n\). They also proved that under the same conditions the same is true for the representation function \(R_{2}(A, n)\) except a few sets. The proofs are elementary and the paper contains some related open problems.