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The Ramanujan Journal

, Volume 46, Issue 3, pp 821–833 | Cite as

Congruences for 7 and 49-regular partitions modulo powers of 7

  • Chandrashekar Adiga
  • Ranganatha Dasappa
Article

Abstract

Let \(b_{k}(n)\) denote the number of k-regular partitions of n. In this paper, we prove Ramanujan-type congruences modulo powers of 7 for \(b_{7}(n)\) and \(b_{49}(n)\). For example, for all \(j\ge 1\) and \(n\ge 0\), we prove that
$$\begin{aligned} b_{7}\Bigg (7^{2j-1}n+\frac{3\cdot 7^{2j-1}-1}{4}\Bigg )\equiv 0\pmod {7^{j}} \end{aligned}$$
and
$$\begin{aligned} b_{49}\Big (7^{j}n+7^{j}-2\Big )\equiv 0\pmod {7^{j}}. \end{aligned}$$

Keywords

Congruences Partitions k-Regular partitions 

Mathematics Subject Classification

05A15 05A17 11P83 

Notes

Acknowledgements

The author wishes to thank the referee for many valuable suggestions and comments. The authors are indebted to Professor Michael Hirschhorn for his valuable suggestions which have substantially improved our paper.

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Copyright information

© Springer Science+Business Media, LLC 2017

Authors and Affiliations

  1. 1.Department of Studies in MathematicsUniversity of MysoreMysoreIndia

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