  A partition of a positive integer $n$ is said to be $t$-core if none of its
hook lengths are divisible by $t$. Recently, two analogues, $\overline{a}_t(n)$
and $\overline{b}_t(n)$, of the $t$-core partition function, $c_t(n)$, have
been introduced by Gireesh, Ray and Shivashankar \cite{grs} and Bandyopadhyay
and Baruah \cite{bb}, respectively. In this article, we prove the lacunarity of
$\overline{b}_t(n)$ modulo arbitrary powers of 2 and 3 for $t=3^\alpha m$ where
$\gcd(m,6)$=1. For a fixed positive integer $k$ and prime numbers $p_i\geq 5$,
we also study the arithmetic density of $\overline{b}_t(n)$ modulo $p_i^k$
where $t=p_1^{a_1}\cdots p_m^{a_m}$. We further prove an infinite family of
congruences for $\overline{b}_3(n)$ modulo arbitrary powers of 2 by employing a
result of Ono and Taguchi on the nilpotency of Hecke operators.
