  For $N$ spin-polarized fermions with harmonic pair interactions in a
$1$-dimensional trap an odd-even effect is found. The spectrum of the
$1$-particle reduced density matrix of the system's ground state differs
qualitatively for $N$ odd and $N$ even. This effect does only occur for strong
attractive and repulsive interactions. Since it does not exists for bosons, it
must originate from the repulsive nature implied by the fermionic exchange
statistics. In contrast to the spectrum, the $1$-particle density and
correlation function for strong attractive interactions do not show any
sensitivity on the number parity. This also suggests that
reduced-density-matrix-functional theory has a more subtle $N$-dependency than
density functional theory.
