  In this paper we produce a few continuations of our previous work on
partitions into fractions. Specifically, we study strictly increasing integer
sequences $\{n_j\}$ such that there are partitions for all integers less than
the floor of $G$, where $G=\frac{n_1}{j} + \frac{n_2}{j} + \cdots +
\frac{n_{j-1}}{j}+\frac{n_j}{j}$, and all summands are distinct terms drawn
from $\frac{n_1}{j} + \frac{n_2}{j} + \cdots +
\frac{n_{j-1}}{j}+\frac{n_j}{j}$. We call such sequences
\enquote{semicomplete}. We find that there are only three semicomplete
arithmetic sequences. We also study sequences that give the maximum number of
pieces that an $M$ dimensional hypercube can be cut into using $N-1$
hyperplanes. We find that these are semicomplete in one, two, three, and four
dimensions. As an aside, we use one of our generating functions to produce what
appears to be a new identity for the Pell constant, a number which is closely
connected to the density of solutions to the negative Pell equation.
