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A partition of $n$ is called a $t$-core partition if none of its hook numberis divisible by $t.$ In 2019, Hirschhorn and Sellers \cite{Hirs2019} obtained aparity result for $3$-core partition function $a_3(n)$. Recently, both authors\cite{MeherJindal2022} proved density results for $a_3(n)$, wherein we provedthat $a_3(n)$ is almost always divisible by arbitrary power of $2$ and $3.$ Inthis article, we prove that for a non-negative integer $\alpha,$ $a_{3^{\alpha}m}(n)$ is almost always divisible by arbitrary power of $2$ and $3.$ Further,we prove that $a_{t}(n)$ is almost always divisible by arbitrary power of$p_i^j,$ where $j$ is a fixed positive integer and $t= p_1^{a_1}p_2^{a_2}\ldotsp_m^{a_m}$ with primes $p_i \geq 5.$ Furthermore, by employing Radu andSeller's approach, we obtain an algorithm and we give alternate proofs ofseveral congruences modulo $3$ and $5$ for $a_{p}(n)$, where $p$ is primenumber. Our results also generalizes the results in \cite{radu2011a}.

Arithmetic density and congruences of $t$-core partitions