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Arithmetic properties of 7-regular partitions

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Abstract

Let \(b_{\ell }(n)\) denote the number of \(\ell \)-regular partitions of n. By employing the modular equation of seventh order, we establish the following congruence for \(b_{7}(n)\) modulo powers of 7: for \(n\ge 0\) and \(j\ge 1\),

$$\begin{aligned} b_{7}\left( 7^{2j-1}n+\frac{3\cdot 7^{2j}-1}{4}\right) \equiv 0 \pmod {7^j}. \end{aligned}$$

We also find some infinite families of congruences modulo 2 and 7 satisfied by \(b_{7}(n)\).

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Acknowledgements

The author thanks the referee for his/her careful reading of the manuscript and helpful suggestions which improved the presentation of this work.

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  1. School of Mathematics and Statistics, Wuhan University, Wuhan, 430072, Hubei, China

    Liuquan Wang

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  1. Liuquan Wang
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Wang, L. Arithmetic properties of 7-regular partitions. Ramanujan J 47, 99–115 (2018). https://doi.org/10.1007/s11139-018-9990-1

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