We present a study of X-ray AGN overdensities in 16 Abell clusters, within the redshift range 0.073<z<0.279, in order to investigate the effect of the hot inter-cluster environment on the triggering of the AGN phenomenon. The X-ray AGN overdensities, with respect to the field expectations, were estimated for sources with L_x>= 10^{42} erg s^{-1} (at the redshift of the clusters) and within an area of 1 h^{-1}_{72} Mpc radius (excluding the core). To investigate the presence or not of a true enhancement of luminous X-ray AGN in the cluster area, we also derived the corresponding optical galaxy overdensities, using a suitable range of $r$-band magnitudes. We always find the latter to be significantly higher (and only in two cases roughly equal) with respect to the corresponding X-ray overdensities. Over the whole cluster sample, the mean X-ray point-source overdensity is a factor of ~4 less than that corresponding to bright optical galaxies, a difference which is significant at a >0.995 level, as indicated by an appropriate t-student test. We conclude that the triggering of luminous X-ray AGN in rich clusters is strongly suppressed. Furthermore, searching for optical SDSS counterparts of all the X-ray sources, associated with our clusters, we found that about half appear to be background QSOs, while others are background and foreground AGN or stars. The true overdensity of X-ray point sources, associated to the clusters, is therefore even smaller than what our statistical approach revealed.
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