A partition of $n$ is called a $t$-core partition if none of its hook number is divisible by $t.$ In 2019, Hirschhorn and Sellers \cite{ Hirs2019} obtained a parity result for $3$-core partition function $a_3(n)$. Recently, both authors \cite{ MeherJindal2022} proved density results for $a_3(n)$, wherein we proved that $a_3(n)$ is almost always divisible by arbitrary power of $2$ and $3.$ In this 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 $t= p_1^{ a_1}p_2^{ a_2}\ldots p_m^{ a_m}$ with primes $p_i \geq 5.$ Furthermore, by employing Radu and Seller's approach, we obtain an algorithm and we give alternate proofs of several congruences modulo $3$ and $5$ for $a_{ p}(n)$, where $p$ is prime number. Our results also generalizes the results in \cite{ radu2011a}.
Categories: math.NT