Some new congruences for overcubic partitions with r-tuples

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2024年11月25日

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Pujashree Buragohain,
Nipen Saikia

摘要

Kim (Ramanujan Math Soc Lect Notes Ser 14:157–163, 2010) introduced the overcubic partition function $\overline{a}(n)$ $\bar{a} (n)$ , which represents the number of all the overlined versions of the cubic partition counted by a(n). Let $ \overline{b}_r(n)$ ${\bar{b}}_{r} (n)$ denote the number of overcubic partitions of n with r-tuples. Several authors established many particular and infinite families of congruences for $ \overline{b}_2(n)$ ${\bar{b}}_{2} (n)$ . In this paper, we show that $ \overline{b}_{2^\beta m+t}(n)\equiv \overline{b}_{t}(n) \,(mod \,2^{\beta +1}), $ ${\bar{b}}_{2^{β} m + t} (n) \equiv {\bar{b}}_{t} (n) (m o d 2^{β + 1}),$ where $\beta \ge 1$ $β \geq 1$ , $m\ge 0$ $m \geq 0$ , and $t\ge 1$ $t \geq 1$ are integers. We also prove some new congruences modulo 8, 16 and 32 for $\overline{b}_{4m+2}(n)$ ${\bar{b}}_{4 m + 2} (n)$ , $\overline{b}_{4m+3}(n)$ ${\bar{b}}_{4 m + 3} (n)$ , $\overline{b}_{8m+2}(n)$ ${\bar{b}}_{8 m + 2} (n)$ , $\overline{b}_{8m+4}(n)$ ${\bar{b}}_{8 m + 4} (n)$ and $\overline{b}_{16m+4}(n)$ ${\bar{b}}_{16 m + 4} (n)$ , where m is any non-negative integer.