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Studies of the mass function (MF) of open clusters of different ages allow usto probe the efficiency with which brown dwarfs evaporate from clusters topopulate the field. Surveys of older clusters (age >100 Myr) are not affectedso severely by several problems encountered in young clusters, such asintra-cluster extinction and large uncertainties in brown dwarf models. Wepresent the results of a deep photometric survey to study the MF of the centralregion 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 anoptical ($riz$ and $Y$-band) photometric survey of Praesepe using the LargeBinocular Telescope Camera covering an area of ~0.59 deg$^2$ in the clustercentre 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, ofwhich 37 are substellar, by comparing with the predictions of a dustyatmosphere model. The MF of those candidates rises from the substellar boundaryuntil ~67 M_Jup and then declines. This is quite different from the forminferred for other open clusters older than 50 Myr, but seems to be similar tothose 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 hadsimilar initial MFs but a different dynamical evolution. Since most of thecandidates are faint, we lack astrometric or spectroscopic follow-ups to testtheir memberships. However, the contaminations by field dwarfs, galaxies, orgiants are found to have little effect on the shape of MF and therefore the MFof `real' cluster members should have similar characteristics.

The substellar mass function in the central region of the open cluster Praesepe from deep LBT observations