A note on the exact formulas for certain $2$-color partitions

Let p≤ 23 be a prime and aₚ(n) count the number of partitions of n using two colors where one of the colors only has parts divisible by p. Using a result of Sussman, we derive the exact formula for aₚ(n) and obtain an asymptotic formula for log aₚ(n). Our results partially extend the work of Mauth, who proved the asymptotic formula for log a₂(n) conjectured by Banerjee et al.