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December 2021 New congruences for overpartitions modulo 3 and 9
Li Zhang
Rocky Mountain J. Math. 51(6): 2269-2273 (December 2021). DOI: 10.1216/rmj.2021.51.2269

Abstract

Let p¯(n) denote the number of overpartitions of n. In this paper, we establish an infinite family of congruences p¯(9α16β3(8n+8i+1))0(mod3), where α,β,n0, is an arbitrary odd prime and 0i< is a nonnegative integer such that (8i+1)=1. In this way, we find various congruences for p¯(n) modulo 3 such as p¯(120n+51,99)0(mod3) and p¯(168n+51,99,123)0(mod3). Furthermore, by studying the generating function of p¯(3n) modulo 9 and applying the fact that ϕ(q)5 is a Hecke eigenform, we obtain another infinite family of congruences p¯(32n)0(mod9), where is a prime with 7(mod9), and n is a nonnegative integer such that (n)=1. This leads to lots of congruences for p¯(n) modulo 9 such as p¯(1029n+441,735,882)0(mod9).

Citation

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Li Zhang. "New congruences for overpartitions modulo 3 and 9." Rocky Mountain J. Math. 51 (6) 2269 - 2273, December 2021. https://doi.org/10.1216/rmj.2021.51.2269

Information

Received: 12 January 2021; Revised: 23 March 2021; Accepted: 12 April 2021; Published: December 2021
First available in Project Euclid: 22 March 2022

Digital Object Identifier: 10.1216/rmj.2021.51.2269

Subjects:
Primary: 05A17
Secondary: 11P83

Keywords: congruence , generating function , overpartition

Rights: Copyright © 2021 Rocky Mountain Mathematics Consortium

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Vol.51 • No. 6 • December 2021
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