Context :

Aims : Hard X-ray , large-area surveys are a fundamental complement of ultra-deep , pencil-beam surveys in obtaining a more complete coverage of the AGN luminosity–redshift plane and finding sizeable samples of “ rare ” AGN .

Methods : We present the results of the photometric and spectroscopic identification of 110 hard X-ray selected sources from 5 additional XMM– Newton fields , nearly doubling the original HELLAS2XMM sample .
Their 2–10 keV fluxes cover the range 6 \times 10 ^ { -15 } -4 \times 10 ^ { -13 } { erg~ { } cm } ^ { -2 } ~ { } { s } ^ { -1 } and the total area surveyed is \sim 0.5 deg ^ { 2 } at the bright flux limit .
We spectroscopically identified 59 new sources , bringing the spectroscopic completeness of the full HELLAS2XMM sample to almost 70 % over a total area of \sim 1.4 deg ^ { 2 } at the bright flux limit .
We found optical counterparts for 214 out of the 232 X-ray sources of the full sample down to R \sim 25 .
We measure the flux and luminosity of the [ OIII ] \lambda 5007 emission line for 59 such sources .

Results : Assuming tha most high X–ray-to-optical flux ratio sources are obscured QSOs , we use the full HELLAS2XMM sample and the CDF samples , to estimate their logN–logS .
We find an obscured QSO surface densities of 50 \pm 23 and 100–400 deg ^ { -2 } down to flux limits of 10 ^ { -14 } and 10 ^ { -15 } { erg~ { } cm } ^ { -2 } ~ { } { s } ^ { -1 } , respectively .
At these flux limits the fraction of X–ray selected obscured QSO turns out to be similar to that of unobscured QSO .
Since X–ray selection misses most Compton thick AGN , the number of obscured QSO may well outnumber that of unobscured QSOs .

We find that hard X–ray selected AGNs with detected [ OIII ] emission span a wide range of L _ { 2 - 10 keV } / L _ { [ OIII ] } with a logarithmic median of ( 2.14 \pm 0.38 ) .
This is marginally higher than that of a sample of optically selected AGNs ( median 1.69 and interquatile range 0.30 ) , suggesting that optically selected samples are at least partly incomplete , and / or [ OIII ] emission is not a perfect isotropic indicator of the nuclear power .
The seven X–ray Bright , Optically Normal Galaxy ( XBONG ) candidates in the sample have L _ { 2 - 10 keV } / L _ { [ OIII ] } { { } _ { > } \atop { } ^ { \sim } } 1000 , while their X-ray and optical luminosities and obscuring column density are similar to those of narrow-line AGNs in the same redshift interval ( 0.075–0.32 ) .
This suggests that while the central engine of narrow-line AGNs and XBONGs looks similar , the narrow-line region in XBONGs could be strongly inhibited or obscured .

Conclusions :