Congruences for an arithmetic function from 3-colored Frobenius partitions

TL;DR

This paper proves new congruences modulo 5 for an arithmetic function derived from 3-colored Frobenius partitions, extending known results and contributing to the understanding of partition congruences.

Contribution

It establishes specific congruences for the function a(n) related to 3-colored Frobenius partitions, expanding the theory of partition congruences.

Findings

a(15n+6) ≡ 0 mod 5

a(15n+12) ≡ 0 mod 5

extends results of Ono on partition congruences

Abstract

Let $a(n)$ defined by $\sum_{n=1}^{\infty}a(n)q^n := \prod_{n=1}^{\infty}\frac{1}{(1-q^{3n})(1-q^n)^3}.$ In this note, we prove that for every non-negative integer $n$, a(15n+6) \equiv 0\pmod{5}, a(15n+12) \equiv 0\pmod{5}. As a corollary, we obtained some results of Ono