Skip to main content
SpringerLink
Log in

An unexpected Ramanujan-type congruence modulo 7 for 4-colored generalized Frobenius partitions

  • Published:
The Ramanujan Journal Aims and scope Submit manuscript

Abstract

In this paper, we present an unexpected Ramanujan-type congruence modulo 7 for \(c\phi _4(n)\), which denotes the number of generalized Frobenius partitions of n with 4 colors. This work extends the recent work of Lin on \(c\phi _4\) modulo 7.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Andrews, G.E.: Generalized Frobenius Partitions. Mem. Am. Math. Soc. 49(301), iv+44 (1984)

    MathSciNet  MATH  Google Scholar 

  2. Baruah, N.D., Sarmah, B.K.: Congruences for generalized Frobenius partitions with \(4\) colors. Discret. Math. 311(17), 1892–1902 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  3. Baruah, N.D., Sarmah, B.K.: Generalized Frobenius partitions with \(6\) colors. Ramanujan J. 38(2), 361–382 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  4. Berndt, B.C.: Ramanujan’s Notebooks (Part III), p. xiv+510. Springer, New York (1991)

    Book  MATH  Google Scholar 

  5. Berndt, B.C.: Number Theory in the Spirit of Ramanujan Student Mathematical Library, vol. 34. American Mathematical Society, Providence (2006)

    MATH  Google Scholar 

  6. Eichhorn, D., Sellers, J.A.: Sellers Computational proofs of congruences for \(2\)-colored Frobenius partitions. Int. J. Math. Math. Sci. 29(6), 333–340 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  7. Garvan, F.G.: Congruences for generalized Frobenius partitions with an arbitrarily large number of colors. Integers 14, Paper No. A7 (2014)

    MathSciNet  MATH  Google Scholar 

  8. Hirschhorn, M.D., Sellers, J. A.: Infinitely many congruences modulo \(5\)-colored Frobenius partitions Ramanujan J. doi:10.1007/s11139-014-9652-x

  9. Kolitsch, L.W.: An extension of a congruence by Andrews for generalized Frobenius partitions. J. Comb. Theory Ser. A 45(1), 31–39 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  10. Kolitsch, L.W.: A Congruence for Generalized Frobenius Partitions with 3 Analytic Number Theory (Allerton Park, 1989), 343C348. Progress in Mathematics, vol. 85. Birkhäuser, Boston (1990)

    Google Scholar 

  11. Lin, B.L.S.: New Ramanujan type congruence modulo \(7\) for \(4\)-colored generalized Frobenius partitions. Int. J. Number Theory 10(3), 637–639 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  12. Lovejoy, J.: Ramanujan-type congruences for three colored Frobenius partitions. J. Number Theory 85(2), 283–290 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  13. Ono, K.: Congruences for Frobenius partitions. J. Number Theory 57(1), 170–180 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  14. Paule, P., Radu, C.S.: The Andrews–Sellers family of partition congruences. Adv. Math. 230(3), 819–838 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  15. Sellers, J.: Congruences involving generalized Frobenius partitions. Int. J. Math. Math. Sci. 16(2), 413–415 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  16. Sellers, J.: New congruences for generalized Frobenius partitions with two or three colors. Discret. Math. 131(1–3), 367–373 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  17. Sellers, J.A.: An unexpected congruence modulo \(5\)-colored generalized Frobenius partitions J. Indian Math. Soc. 2013, Special volume to commemorate the 125th birth anniversary of Srinivasa Ramanujan, pp. 97–103

  18. Xia, E.X.W.: A Ramanujan-type congruence modulo \(5\)-colored generalized Frobenius partitions Ramanujan J. doi:10.1007/s11139-014-9642-z

  19. Xia, E.X.W.: Proof of a conjecture of Baruah and Sarmah on generalized Frobenius partitions with \(6\) colors. J. Number Theory 147, 852–860 (2015)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgments

The authors are very grateful to the anonymous referee for his/her careful reading and valuable comments and suggestions.

Author information

Authors and Affiliations

  1. School of Mathematical Sciences, Dalian University of Technology, Dalian, 116024, People’s Republic of China

    Wenlong Zhang

  2. Department of Mathematics, East China Normal University, 500 Dongchuan Road, Shanghai, 200241, People’s Republic of China

    Chun Wang

Authors
  1. Wenlong Zhang
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Chun Wang
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Chun Wang.

Additional information

This work was supported by the Fundamental Research Funds for the Central Universities.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Zhang, W., Wang, C. An unexpected Ramanujan-type congruence modulo 7 for 4-colored generalized Frobenius partitions. Ramanujan J 44, 125–131 (2017). https://doi.org/10.1007/s11139-015-9770-0

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11139-015-9770-0

Keywords

Mathematics Subject Classification

Search

Navigation