For integers k and l with \(1\le k\le l\) , let \(a_{k,l}(n)\) be the number of partitions of n where each part can appear with three colors such that parts divisible by k may take either the first or second color, parts divisible by l may take either the first or third color, and parts divisible by both k and l may take any of the three colors. We apply the result of Sussman to derive the asymptotic formulas for \(\log a_{1,p}(n)\) and \(\log a_{p,p}(n)\) when \(p\le 11\) is a prime and \(\log a_{p,r}(n)\) when \((p,r)\in \{(2,3),(2,5),(2,7),(3,5)\}\) .
