  We present the deformed (for the presence of Calogero potential terms)
one-dimensional quantum oscillator with the exceptional Lie superalgebra $F(4)$
as spectrum-generating superconformal algebra. The Hilbert space is given by a
$16$-ple of square-integrable functions. The energy levels are $\frac{2}{3}+n$,
with $n=0,1,2,\ldots$. The ground state is $7$ times degenerate. The excited
states are $8$ times degenerate. The $(7,8,8,8,\ldots )$ semi-infinite tower of
states is recovered from the $(7;8;1)$ supermultiplet of the ${\cal N}=8$
worldline supersymmetry. The model is unique, up to similarity transformations,
and admits an octonionic-covariant formulation which manifests itself as
"quasi-nonassociativity". This means, in particular, that the Calogero coupling
constants are expressed in terms of the octonionic structure constants. The
associated $F(4)$ superconformal quantum mechanics is also presented.
