Congruences Modulo Powers of 7 for the Reciprocal Crank Parity Function

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 $a(7n+2)\equiv 0(\mathrm{mod}7)$ for all $n\ge 0$. We prove new congruences for the reciprocal crank parity function modulo powers of $7$.