Distribution of Andrews' Singular Overpartitions $\overline{C}_{p,1}(n)$

Andrews introduced the partition function $\overline{C}_{k, i}(n)$, called singular overpartition, which counts the number of overpartitions of $n$ in which no part is divisible by $k$ and only parts $\equiv \pm i\pmod{k}$ may be overlined. We study the parity and distribution results for $\overline{C}_{k,i}(n),$ where $k>3$ and $1\leq i \leq \left\lfloor\frac{k}{2}\right\rfloor$. More particularly, we prove that for each integer $\ell\geq 2$ depending on $k$ and $i$, the interval $\left[\ell, \frac{\ell(3\ell+1)}{2}\right]$ $\Big($resp.\ $\left[2\ell-1, \frac{\ell(3\ell-1)}{2}\right] \Big)$ contains an integer $n$ such that $\overline{C}_{k,i}(n)$ is even (resp.\ odd). Finally we study the distribution for $\overline{C}_{p,1}(n)$ where $p\geq 5$ be a prime number.