  Studies of the mass function (MF) of open clusters of different ages allow us
to probe the efficiency with which brown dwarfs evaporate from clusters to
populate the field. Surveys of older clusters (age >100 Myr) are not affected
so severely by several problems encountered in young clusters, such as
intra-cluster extinction and large uncertainties in brown dwarf models. We
present the results of a deep photometric survey to study the MF of the central
region of the old open cluster Praesepe (age~590$^{+150}_{-120}$ Myr, distance
~190$^{+6.0}_{-5.8}$ pc), down to the substellar regime. We performed an
optical ($riz$ and $Y$-band) photometric survey of Praesepe using the Large
Binocular Telescope Camera covering an area of ~0.59 deg$^2$ in the cluster
centre from $i$~19.0 mag (~100 M_Jup) down to a 5$\sigma$ detection limit at
$i$~25.6mag (~40 M_Jup). The survey is approximately 95% complete at $i=23.8$
mag and $z=22.0$ mag (~55 M_Jup). We identify 59 cluster member candidates, of
which 37 are substellar, by comparing with the predictions of a dusty
atmosphere model. The MF of those candidates rises from the substellar boundary
until ~67 M_Jup and then declines. This is quite different from the form
inferred for other open clusters older than 50 Myr, but seems to be similar to
those found in very young open clusters, the MFs of which peak at ~10 M_Jup.
Either Praesepe really does have a different MF from other clusters or they had
similar initial MFs but a different dynamical evolution. Since most of the
candidates are faint, we lack astrometric or spectroscopic follow-ups to test
their memberships. However, the contaminations by field dwarfs, galaxies, or
giants are found to have little effect on the shape of MF and therefore the MF
of `real' cluster members should have similar characteristics.
