Is it true that every sufficiently large positive integer can be written as a sum of a square free number and a perfect square ?

Question

Is it true that $\exists k \in \mathbb Z^+$ such that every integer $n >k$ can be written as a sum of a square free number and a perfect square ?

Answer 1

Yes, it is true. It was proved for first by Estermann, and in the following Mirsky's article you can find good asymptotics on the number of representations, that is obviously given by $$ R(n)=\sum_{k=0}^{\lfloor\sqrt{n}\rfloor}\mu^2(n-k^2).$$