Distribution of Andrews’ singular overpartitions \({\overline{C}}_{p,1}(n)\)

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