Congruences Modulo Powers of 7 for the Reciprocal Crank Parity Function

TL;DR

This paper establishes new congruences modulo powers of 7 for the reciprocal crank parity function, which counts certain colored integer partitions, extending previous results on its divisibility properties.

Contribution

It introduces novel congruences modulo powers of 7 for the reciprocal crank parity function, expanding understanding of its arithmetic properties.

Findings

Proves congruences modulo powers of 7 for the reciprocal crank parity function.

Extends previous divisibility results to higher powers of 7.

Provides new insights into the arithmetic structure of the sequence.

Abstract

Amdeberhan and Merca recently studied arithmetic properties of the sequence $a(n)$, the reciprocal of the crank parity function, which counts the number of integer partitions of weight $n$ whose even parts are monochromatic and whose odd parts may appear in one of three colors (OEIS A298311). A key result of their work was the congruence $a(7n + 2) \equiv 0 \pmod{7}$ for all $n \geq 0$. We prove new congruences for the reciprocal crank parity function modulo powers of $7$.