Partitions with multiplicities associated with divisor functions

We consider colored partitions of a positive integer n, where the number of times a particular colored part m may appear in a partition of n is equal to the sum of the powers of the divisors of m. An asymptotic formula is derived for the number of such partitions. We also derive an asymptotic formula for the number of partitions of n into c colors. In order to achieve the desired bounds on the minor arcs arising from the Hardy-Littlewood circle method, we generalize a bound on an exponential sum twisted by a generalized divisor function due to Motohashi.