  MATEMATIČKI VESNIK Vol. 70, No. 1, pp. 55–63 (2018)

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## New congruences for overcubic partition function

### C. Shivashankar and M. S. Mahadeva Naika

Department of Mathematics, Reva University, Rukmini Knowledge Park, Yelahanka, Bengaluru – 560 064, Karnataka, India E-mail: shankars224@gmail.com and Department of Mathematics, Bangalore University, Central College Campus, Bangalore – 560 001, Karnataka, India E-mail: msmnaika@rediffmail.com

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 $\overline{a}\left(3n+2\right)\equiv 0\phantom{\rule{4.44443pt}{0ex}}\left(mod\phantom{\rule{0.277778em}{0ex}}3\right)$. 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 $n\ge 0$, $\overline{a}\left(8n+5\right)\equiv 0\phantom{\rule{4.44443pt}{0ex}}\left(mod\phantom{\rule{0.277778em}{0ex}}16\right)$, $\overline{a}\left(8n+7\right)\equiv$ 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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