First we show, from the BBGKY hierarchy and under exponential clustering assumption, that the three-dimensional one-component plasma obeys the compressibility sum rule which links the compressibility to the fourth moment of the two-point correlation function. Then it is proved that the first equation of the hierarchy is equivalent to the known value of the correlation function of momentum current and density. Finally we are concerned with the energy density, its definition, and its correlation with the numerical density and with itself.