Divisibility properties of weighted k regular partitions

We study a generalized class of weighted k-regular partitions defined by \[ ∑ₙ₌₀∞ ck, r_1, r_2(n) q^n = ∏ₙ₌₁∞ {(1 - qⁿᵏ)r_1}{(1 - q^n)r_2}, \] which extends the classical k-regular partition function bₖ(n). We establish new infinite families of Ramanujan-type congruences, divisibility results, and positive-density prime sets for which ck, r₁, r₂(n) vanishes modulo a given prime. These results generalize recent work on $5$-regular partitions and reveal deeper modular and combinatorial structures underlying weighted partition functions.