# Divisibility properties of weighted $k$ regular partitions

## TL;DR

This paper explores a generalized class of weighted k-regular partitions, establishing new Ramanujan-type congruences, divisibility results, and prime sets where the partition function vanishes modulo primes.

## Contribution

It introduces a broad generalization of k-regular partitions and proves novel congruences and divisibility properties extending classical results.

## Key findings

- Established new Ramanujan-type congruences
- Identified divisibility results for the partition functions
- Found prime sets with vanishing partition values modulo primes

## Abstract

We study a generalized class of weighted $k$-regular partitions defined by \[ \sum_{n=0}^{\infty} c_{k, r_1, r_2}(n) q^n = \prod_{n=1}^{\infty} \frac{(1 - q^{nk})^{r_1}}{(1 - q^n)^{r_2}}, \] which extends the classical $k$-regular partition function $b_k(n)$. We establish new infinite families of Ramanujan-type congruences, divisibility results, and positive-density prime sets for which $c_{k, r_1, r_2}(n)$ vanishes modulo a given prime.