Congruences modulo powers of 3 for generalized Frobenius partitions with six colors
In his 1984 AMS Memoir, Andrews introduced the $k$-colored generalized Frobenius partition function. Let $c\phi _k(n)$ denote the number of generalized Frobenius partitions of $n$ with $k$ colors. Recently, congruences modulo 4, 9 and 27 for $c\phi _6(n)$ were proved by Baruah and Sarmah, Hirschhorn, and Xia. In this paper, we prove several congruences modulo powers of 3 for $c\phi _6(n)$ by using the generating function for $c\phi _6(3n+1)$ due to Hirschhorn. In particular, we confirm a conjecture on a congruence modulo 243 for $c\phi _6(n)$.