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New infinite families of congruences modulo 3, 5 and 7 for overpartition function. (English) Zbl 1441.11267

Summary: Let \(\bar{p}(n)\) denote the number of overpartitions of a non-negative integer \(n\). In this paper, we prove two new infinite families of congruences modulo 3 for \(\bar{p}(n)\) by using Ramanujan’s theta-function identities. Particularly, we prove that, for any integer \(\alpha \geq 0, \bar{p} (9^{\alpha +1}(24n+23)) \equiv 0 \pmod 3\) and \(\bar{p} (9^{\alpha +1}(24n+22)+1) \equiv 0 \pmod 3\). Furthermore, we prove some new congruences modulo 5 and 7 for \(\bar{p}(n)\). For example, we prove that \(\bar{p}(5n+k+3) \equiv 0 \pmod 5\), where \(k = 3n^2 \pm n\).

MSC:

11P83 Partitions; congruences and congruential restrictions
05A15 Exact enumeration problems, generating functions
05A17 Combinatorial aspects of partitions of integers
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