Harmonic partitions: partitions with given sum of the reciprocals of the parts (Q353314)

The authors prove: every integer \(n \geq 24\) can be partitioned into positive integers, \(n=a_1+ \cdots + a_t\), where in addition \(\frac{1}{a_1}+ \cdots + \frac{1}{a_n}=1\) holds. For \(n \geq 78\) this is possible with distinct \(a_i\). The latter is a result of \textit{R. L. Graham} [J. Aust. Math. Soc. 3, 435--441 (1963; Zbl 0142.01304)]. In both cases the authors also classify the small values \(n <24\) (or \(n<78\)) that have a solution. The proofs are elementary.