Partitions with prescribed sum of reciprocals: computational results

Abstract

For a positive rational $\alpha$, call a set of distinct positive integers $\{a_1, a_2, \ldots, a_r\}$ an $\alpha$-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 $\alpha$. Define $n_{\alpha}$ to be the smallest positive integer such that for all $n \ge n_{\alpha}$ an $\alpha$-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 $\alpha$ such that $n_{\alpha} \le 100$.