Partitions with prescribed sum of reciprocals: asymptotic bounds

In $1963$ Graham proved that every positive integer $n \ge 78$ can be written as a sum of distinct positive integers $a_1, a_2, \ldots, a_r$ for which $\frac{1}{a_1} + \frac{1}{a_2} + \ldots + \frac{1}{a_r}$ is equal to $1$. In the same paper he managed to further generalize this, and showed that for all positive rationals $\alpha$ and all positive integers $m$, there exists an $n_{\alpha, m}$ such that every positive integer $n \ge n_{\alpha, m}$ has a partition with distinct parts, all larger than or equal to $m$, and such that the sum of reciprocals is equal to $\alpha$. No attempt was made to estimate the quantity $n_{\alpha, m}$, however. With $n_{\alpha} := n_{\alpha, 1}$, in this paper we provide near-optimal upper bounds on $n_{\alpha}$ and $n_{\alpha, m}$, as well as bounds on the cardinality of the set $\{\alpha : n_{\alpha} \le n\}$.