Arithmetic Identities for Some Analogs of $5$-core Partition Function

Recently, Gireesh, Ray, and Shivashankar studied an analog, ${\bar{a}}_t(n)$, of the $t$-core partition function, $c_t(n)$. In this paper, we study the function ${\bar{a}}_5(n)$ in conjunction with $c_5(n)$ as well as another analogous function ${\bar{b}}_5(n)$. We also find several arithmetic identities for ${\bar{a}}_5(n)$ and ${\bar{b}}_5(n)$.