Note on a theorem of Birch and Erdős

Let $p,q>1$ be two relatively prime integers and $\mathbb{N}$ the set of nonnegative integers. Let $f_{p,q}(n)$ be the number of different expressions of $n$ written as a sum of distinct terms taken from $\{p^{\alpha }{q}^{\beta }:\alpha ,\beta \in \mathbb{N}\}$. Erd\H os conjectured and then Birch proved that $f_{p,q}(n)\ge 1$ provided that $n$ is sufficiently large. In this note, for all sufficiently large number $n$ we prove $$f_{p,q}(n)=2^{\frac{(\log n{)}^2}{2\log p\log q}(1+O(\log \log n\mathrm{/}\log n))}\mathrm{.}$$ We also show that ${\lim }_{n\to \mathrm{\infty }}{f}_{2,q}(n+1)\mathrm{/}{f}_{2,q}(n)=1.$ Additionally, we will point out the relations between $f_{2,q}(n)$ and $m$-ary partitions.