Congruences modulo powers of 3 for generalized Frobenius partitions with six colors (Q2833601)

Generalized Frobenius partitions of \(n\) into \(k\) colors were first introduced by \textit{G. E. Andrews} [Mem. Am. Math. Soc. 301, 44 p. (1984; Zbl 0544.10010)]. Let \(c\phi_k(n)\) denote the number of \(k\)-colored generalized Frobenius partitions of \(n\). Using the generating function for \(c\phi_6(3n+1)\) obtained by \textit{M. D. Hirschhorn} [Ramanujan J. 40, No. 3, 463--471 (2016; Zbl 1407.11120)], the authors prove several congruences modulo powers of \(3\) for \(c\phi_6(n)\). In particular, they show that NEWLINE\[NEWLINEc\phi_6(27n+16)\equiv 0\pmod{3^5},NEWLINE\]NEWLINE a result conjectured by \textit{E. X. W. Xia} [J. Number Theory 147, 852--860 (2015; Zbl 1386.11123)].