Abstract
Let \(\bar{p}(n)\) denote the number of overpartitions of n. Recently, numerous congruences modulo powers of 2, 3 and 5 were established regarding \(\bar{p}(n)\). In particular, Xia discovered several infinite families of congruences modulo 9 and 27 for \(\bar{p}(n)\). Moreover, Xia conjectured that for \(n\ge 0\), \(\bar{p}(96n+76) \equiv 0\ (\mathrm{mod}\ 243)\). In this paper, we confirm this conjecture by using theta function identities and the (p, k)-parametrization of theta functions.
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Acknowledgements
The authors would like to thank the anonymous referee for valuable corrections and comments. This work was supported by Jiangsu National Funds for Distinguished Young Scientists (Grant No. BK20180044) and Chinese Postdoctoral Science Foundation (Grant No. 2018T110444).
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Huang, X., Yao, O.X.M. Proof of a conjecture on a congruence modulo 243 for overpartitions. Period Math Hung 79, 227–235 (2019). https://doi.org/10.1007/s10998-019-00283-4
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DOI: https://doi.org/10.1007/s10998-019-00283-4