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Arithmetic properties of overpartition triples

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Abstract

Let \({\overline p _3}\left( n \right)\) be the number of overpartition triples of n. By elementary series manipulations, we establish some congruences for \({\overline p _3}\left( n \right)\) modulo small powers of 2, such as

$${\overline p _3}\left( {16n + 14} \right) \equiv 0\left( {\bmod 32} \right),{\overline p _3}\left( {8n + 7} \right) \equiv 0\left( {\bmod 64} \right)$$

. We also find many arithmetic properties for \({\overline p _3}\left( n \right)\) modulo 7, 9 and 11, involving the following infinite families of Ramanujan-type congruences: for any integers α ≥ 1 and n ≥ 0, we

$$\bar p_3 \left( {3^{2\alpha + 1} \left( {3n + 2} \right)} \right) \equiv 0\left( {\bmod 9\cdot2^4 } \right),\bar p_3 \left( {4^{2\alpha - 1} \left( {56n + 49} \right)} \right) \equiv 0\left( {\bmod 7} \right),\bar p_3 \left( {7^{2\alpha + 1} \left( {7n + 3} \right)} \right) \equiv \bar p_3 \left( {7^{2\alpha + 1} \left( {7n + 5} \right)} \right) \equiv \bar p_3 \left( {7^{2\alpha + 1} \left( {7n + 6} \right)} \right) \equiv 0\left( {\bmod 7} \right)$$

, and for r ∈ {1, 2, 3, 4, 5, 6}, \({\overline p _3}\left( {11 \cdot {7^{4\alpha - 1}}\left( {7n + r} \right)} \right) \equiv 0\left( {\bmod 11} \right)\).

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  1. Department of Mathematics, National University of Singapore, Singapore, 119076, Singapore

    Liu Quan Wang

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  1. Liu Quan Wang
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Wang, L.Q. Arithmetic properties of overpartition triples. Acta. Math. Sin.-English Ser. 33, 37–50 (2017). https://doi.org/10.1007/s10114-016-5673-2

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  • DOI: https://doi.org/10.1007/s10114-016-5673-2

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