Some new congruences for overcubic partitions with r-tuples

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