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

Volume 175 / 2016

Chao Gu, Liuquan Wang, Ernest X. W. Xia Acta Arithmetica 175 (2016), 291-300 MSC: Primary 05A17; Secondary 11P83. DOI: 10.4064/aa8526-7-2016 Published online: 15 September 2016

Abstract

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 $c\phi _6(n)$.

Authors

  • Chao GuDepartment of Mathematics
    Jiangsu University
    Zhenjiang, Jiangsu 212013, P.R. China
    e-mail
  • Liuquan WangDepartment of Mathematics
    National University of Singapore
    10 Lower Kent Ridge Road
    Singapore, 119076, Singapore
    e-mail
  • Ernest X. W. XiaDepartment of Mathematics
    Jiangsu University
    Zhenjiang, Jiangsu 212013, P.R. China
    e-mail

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