Some new congruences for overcubic partitions with r-tuples

Abstract

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