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The Ramanujan Journal

, Volume 47, Issue 2, pp 435–445 | Cite as

Some congruences modulo 5 and 25 for overpartitions

  • Shane Chern
  • Manosij Ghosh Dastidar
Article
  • 278 Downloads

Abstract

We present two new Ramanujan-type congruences modulo 5 for overpartitions. We also give an affirmative answer to a conjecture of Dou and Lin, which includes four congruences modulo 25 for overpartition.

Keywords

Overpartitions Ramanujan-type congruence Ramanujan’s theta function 

Mathematics Subject Classification

11P83 05A17 

Notes

Acknowledgements

We thank George E. Andrews and Yucheng Liu for some helpful discussions. We also want to thank the referee for useful comments.

References

  1. 1.
    Alaca, Ş., Williams, K.S.: The number of representations of a positive integer by certain octonary quadratic forms. Funct. Approx. Comment. Math. 43(1), 45–54 (2010)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Berndt, B.C.: Ramanujan’s Notebooks. Part III, p. xiv+510. Springer, New York, (1991)CrossRefGoogle Scholar
  3. 3.
    Berndt, B.C.: Ramanujan’s Notebooks. Part V, p. xiv+624. Springer, New York (1998)CrossRefGoogle Scholar
  4. 4.
    Chen, W.Y.C., Sun, L.H., Wang, R.H., Zhang, L.: Ramanujan-type congruences for overpartitions modulo 5. J. Number Theory 148, 62–72 (2015)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Chen, W.Y.C., Xia, E.X.W.: Proof of a conjecture of Hirschhorn and Sellers on overpartitions. Acta Arith. 163(1), 59–69 (2014)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Dou, D.Q.J., Lin, B.L.S.: New Ramanujan type congruences modulo \(5\) for overpartitions. Ramanujan J. (2016).  https://doi.org/10.1007/s11139-016-9782-4 MathSciNetCrossRefGoogle Scholar
  7. 7.
    Hirschhorn, M.D.: Some congruences for overpartitions. N. Z. J. Math. 46, 141–144 (2016)MathSciNetzbMATHGoogle Scholar
  8. 8.
    Hirschhorn, M.D., Sellers, J.A.: An infinite family of overpartition congruences modulo \(12\). Integers. 5(1), 20 (2005)Google Scholar
  9. 9.
    Kim, B.: The overpartition function modulo \(128\). Integers 8, 38 (2008)Google Scholar
  10. 10.
    Mahlburg, K.: The overpartition function modulo small powers of \(2\). Discret. Math. 286(3), 263–267 (2004)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Radu, S., Sellers, J.A.: Congruence properties modulo \(5\) and \(7\) for the \({{\rm pod}}\) function. Int. J. Number Theory 7(8), 2249–2259 (2011)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Xia, E.X.W.: Congruences modulo \(9\) and \(27\) for overpartitions. Ramanujan J. 42(2), 301–323 (2017)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Yao, O.X.M., Xia, E.X.W.: New Ramanujan-like congruences modulo powers of \(2\) and \(3\) for overpartitions. J. Number Theory 133(6), 1932–1949 (2013)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2017

Authors and Affiliations

  1. 1.Department of MathematicsPennsylvania State UniversityPAUSA
  2. 2.Department of Mathematical SciencesPondicherry UniversityKalapetIndia

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