New congruences for overcubic partition function

C. Shivashankar and M. S. Mahadeva Naika

Abstract: In 2010, Byungchan Kim introduced a new class of partition function $\overline{a}\left(n\right)$, the number of overcubic partitions of $n$ and established $\overline{a}(3n+2)\equiv 0(mod3)$. Our goal is to consider this function from an arithmetic point of view in the spirit of Ramanujan’s congruences for the unrestricted partition function $p\left(n\right)$. We prove a number of results for $\overline{a}\left(n\right)$, for example, for $\alpha \ge 0$ and $n\ge 0$, $\overline{a}(8n+5)\equiv 0(mod16)$, $\overline{a}(8n+7)\equiv$ 0(32)32$,$a(8 32+2n+32+2)3 a(8n+1)(8)8$·$

Keywords: Overcubic partitions; congruences; theta function.

Classification (MSC2000): 11P83; 05A15, 05A17

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