On the $r$-th Root Partition Function

Abstract

The well known partition function $p(n)$ has a long research history, where $p(n)$ denotes the number of solutions of the equation $n = a_1 + \cdots + a_k$ with integers $1 \leq a_1 \leq \cdots \leq a_k$. In this paper, we investigate a new partition function. For any real number $r \gt 1$, let $p_r(n)$ be the number of solutions of the equation $n = \lfloor \sqrt[r]{a_1} \rfloor + \cdots + \lfloor \sqrt[r]{a_k} \rfloor$ with integers $1 \leq a_1 \leq \cdots \leq a_k$, where $\lfloor x \rfloor$ denotes the greatest integer not exceeding $x$. In this paper, it is proved that $\exp(c_1 n^{r/(r+1)}) \leq p_r(n) \leq \exp(c_2n^{r/(r+1)})$ for two positive constants $c_1$ and $c_2$ (depending only $r$).