Chan studied the cubic partition function a(n), the number of partitions of a positive integer n where the even parts appear in two colors. Also, he obtained an infinite family of congruences modulo large powers of 3 for a(n), which are analogues to the Ramanujan-type congruences. Let \(\overline{a}_\ell (n)\) denote the number of \(\ell \)-regular cubic partitions of n where the first occurrence of each distinct odd part may be overlined. In this paper, we have established many infinite family of congruences modulo powers of 2 and 3 for \(\overline{a}_3(n)\) and modulo powers of 2 for \(\overline{a}_5(n)\). For example, for all \(n \ge 0\) and \(\alpha , \beta \ge 0\),

where \(c_1 \in \{11, 17, 23, 29\}\).

Chan 研究了三次分区函数a ( n )，即正整数n的分区数，其中偶数部分以两种颜色出现。此外，他获得了一个无穷大的同余族，对a ( n ) 取模 3 的大幂，这类似于拉马努金型同余。让\(\overline{a}_\ell (n)\)表示\(\ell \) - n 的正则三次分区的数量，其中每个不同奇数部分的第一次出现可能被加上划线。在本文中，我们为\(\overline{a}_3(n)\)建立了许多同余模2 和3 的模幂族和2 的模幂\(\overline{a}_5(n)\)。例如，对于所有\(n \ge 0\)和\(\alpha , \beta \ge 0\)，

其中\(c_1 \in \{11, 17, 23, 29\}\)。