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 $a_3(n)$. Motivated by this result, both the authors \cite{ MeherJindal2022} recently proved that for a non-negative integer $\alpha,$ $a_{ 3^{ \alpha} m}(n)$ is almost always divisible by arbitrary power of $2$ and $3$ and $a_{ t}(n)$ 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 $p_i \geq 5.$ In this article, by using Hecke eigenform theory, we obtain infinite families of congruences and multiplicative identities for $a_2(n)$ and $a_{ 13}(n)$ modulo $2$ which generalizes some results of Das \cite{ Das2016}.
Categories: math.NT