Ramanujan-Type congruences for cubic partition functions

TL;DR

This paper establishes Ramanujan-type congruences for the cubic partition function, showing specific divisibility properties for its values at certain arithmetic progressions, extending previous results and suggesting deeper modular structure.

Contribution

It generalizes known results by proving new Ramanujan-type congruences for the cubic partition function, revealing its arithmetic properties.

Findings

Proves that a(5^4n+547) is divisible by 25.

Shows that a(7^3n+190) is divisible by 49.

Establishes that a(7^3n+288) and a(7^3n+337) are divisible by 49.

Abstract

The cubic partitions of a natural number $n$, introduced by Chan and Kim, have generating function $\sum_{n=0}^{\infty}a(n)q^n= \frac{1}{(q; q)_{\infty}(q^2; q^2)_{\infty}}.$ In this paper, we generalize some results of Chen-Lin, which suggest that $a(n)$ should have analogous properties of the ordinary partition function. Specifically, we show that for every non-negative integer $n$, $a(5^4n+547)\equiv 0\pmod{5^2}, a(7^3n+190)\equiv 0\pmod{7^2}, a(7^3n+288 \equiv 0\pmod{7^2} and a(7^3n+337)\equiv 0\pmod{7^2}.$