The Ramanujan Journal

, Volume 46, Issue 1, pp 1–18 | Cite as

Congruences modulo 64 and 1024 for overpartitions

  • Olivia X. M. Yao


Recently, several infinite families of congruences modulo 32, 64 and 256 for \(\overline{p}(n)\) have been established by Yang et al. where \(\overline{p} (n)\) denotes the number of overpartitions of n. In this paper, we establish congruences modulo 64 and 1024 by using identities for \(r_3(n)\) and \(r_7(n)\), where \(r_k(n)\) is the number of representations of n as a sum of k squares. For example, we prove that for \(n,\ \alpha \ge 0\),
$$\begin{aligned} \overline{p}( 3^{16\alpha +15}(24n+5) ) \equiv \overline{p}( 3^{16\alpha +15}(24n+13) ) \equiv 0\ (\mathrm{mod}\ 1024). \end{aligned}$$
In particular, we generalize some congruences for \(\overline{p}(n) \) due to Yang et al.


Overpartitions Congruences Sum of squares 

Mathematics Subject Classification

11P83 05A17 



The author would like to thank the anonymous referee for valuable corrections and comments.


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Copyright information

© Springer Science+Business Media New York 2016

Authors and Affiliations

  1. 1.Department of MathematicsJiangsu UniversityZhenjiangPeople’s Republic of China

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