Partitions with prescribed sum of reciprocals: computational results

For a positive rational $α$, call a set of distinct positive integers $\{a_1, a_2, \ldots, a_r\}$ an $α$-partition of $n$, if the sum of the $a_i$ is equal to $n$ and the sum of the reciprocals of the $a_i$ is equal to $α$. Define $n_α$ to be the smallest positive integer such that for all $n \ge n_α$ an $α$-partition of $n$ exists and, for a positive integer $M \ge 2$, define $N_M$ to be the smallest positive integer such that for all $n \ge N_M$ a $1$-partition of $n$ exists where $M$ does not divide any of the $a_i$. In this paper we determine $N_M$ for all $M \ge 2$, and find the set of all $α$ such that $n_α \le 100$.