Congruences modulo powers of 3 for 6-colored generalized Frobenius partitions

Abstract

In his 1984 AMS Memoir, Andrews introduced the family of functions $c\phi_k(n)$, which denotes the number of $k$-colored generalized Frobenius partitions of $n$. In this paper, we prove three congruences and three internal congruences modulo powers of 3 for $c\phi_6(n)$ by utilizing the generating function of $c\phi_6(3n+1)$ due to Hirschhorn. Finally, we conjecture two families of congruences and two families of internal congruences modulo arbitrary powers of 3 for $c\phi_6(n)$, which strengthen a conjecture due to Gu, Wang and Xia in 2016