How many of the partitions of a natural number $n$ are comprised only of its divisors? That is, if $$p(n)=\sum_{\sum_{1}^n kj_k=n:j_k\geq 0} 1_{\[j_1,j_2,...\]},$$ is the ordinary partition function (i.e. the total number of partitions of $n$), then I want to know something about the counting function $$s(n)=\sum_{\sum_{d|n}dj_d=n:j_d\geq 0}1_{[j_1,...,j_d,...]}.$$

I would be happy to hear of anything that is known about this function, but I am particularly interested in (a) its generating functions, and (b) a bijection between this restricted partition and another (hopefully more intuitive to count) restricted partition. Any insights would be welcome.

I should add that google finds a number of papers that study "partitions of $n$ into divisors of $m$", e.g. Gupta, 1970s, but those methods reduce to rather vacuous statements when evaluated at $m=n$.

Thanks!