  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', \textit{J. Integer Seq.} \textbf{27} (2024), \#
24.4.5] proved new arithmetic identities satisfied by $\overline{a}_5(n)$. In
this article, we study the arithmetic densities of $\overline{a}_t(n)$ 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 $\overline{a}_3(n)$ modulo
arbitrary powers of 2.
