Abstract

In 2010, Byungchan Kim introduced a new class of partition function $\overline{a}(n)$, the number of overcubic partitions of $n$ and established $\overline{a}(3n+2)\equiv 0\pmod{3}$. 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 $p(n)$. We prove a number of results for $\overline{a}(n)$, for example, for $\alpha \ge 0$ and $n \ge 0$, $\overline{a}(8n+5)\equiv 0\pmod{16}$, $\overline{a}(8n+7)\equiv 0\pmod{32}$, $\overline{a}(8\cdot 3^{2\alpha+2}n+3^{2\alpha+2})\equiv 3^{\alpha} \overline{a}(8n+1)\pmod{8}$.

Keywords: Overcubic partitions; congruences; theta function.

MSC: 11P83, 05A15, 05A17

Pages:  55$-$63     

Volume  70 ,  Issue  1 ,  2018