Arithmetic properties of an analogue of $t$-core partitions

An integer partition of a positive integer $n$ is called to be $t$-core if none of its hook lengths are divisible by $t$. Recently, Gireesh, Ray and Shivashankar [A new analogue of $t$-core partitions', \textit{Acta Arith.} \textbf{199} (2021), 33-53] introduced an analogue $\overline{a}_t(n)$ of the $t$-core partition function $c_t(n)$. They obtained certain multiplicative formulas and arithmetic identities for $\overline{a}_t(n)$ where $t \in \{3,4,5,8\}$ and studied the arithmetic density of $\overline{a}_t(n)$ modulo $p_i^{j}$ where $t=p_1^{a_1}\cdots p_m^{a_m}$ and $p_i\geq 5$ are primes. Very recently, Bandyopadhyay and Baruah [Arithmetic identities for some analogs of the 5-core partition function’, $\mathit{\text{J.}}\mathit{\text{Integer}}\mathit{\text{Seq.}}$ $\mathbf{\text{27}}$ (2024), # 24.4.5] proved new arithmetic identities satisfied by ${\bar{a}}_5(n)$. In this article, we study the arithmetic densities of ${\bar{a}}_t(n)$ modulo arbitrary powers of 2 and 3 for $t=3^{\alpha }m$ where $\gcd (m,6)$=1. Also, employing a result of Ono and Taguchi on the nilpotency of Hecke operators, we prove an infinite family of congruences for ${\bar{a}}_3(n)$ modulo arbitrary powers of 2.