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The Ramanujan Journal

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

Congruences modulo 64 and 1024 for overpartitions

  • Olivia X. M. Yao
Article

Abstract

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.

Keywords

Overpartitions Congruences Sum of squares 

Mathematics Subject Classification

11P83 05A17 

Notes

Acknowledgments

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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