IMHO
As I recall from investagating long ago the running HP is the HP at the compressor to deliver the cfm to replinish the tank. The motor RPM, say 3600 is stepped down to say 600 rpm.
A 1 HP motor would be delivering 6 HP to the compressor. This is of course for the belt driven compressors
Then the motor is slightly larger to ensure it isn't running at full load with the losses. I always measure to see what is needed for a power circuit. It is misleading.
I understand about the law of conservation of energy.
The Hp is calculated
Horsepower = Torque X RPM/5252 What I explained poorly is that is how someone states (incorectly) the HP for thier advertising.
Conservation of energy dictates that you cannot get more pneumatic power out of a compressor than the electrical power you put into it. A compressor simply converts electrical power into pneumatic power, less losses. A motor is the same in that it converts electrical power into mechanical power, less losses.
I must not have made my point clearly. I don't believe that the compressor manufacturer's are listing a "horsepower" that is an actual expression of real "power," either mechanical or electrical power.
Instead, I believe that they are using the term "horsepower" as an indication of the "time" that it would take to charge or recover the compressor tank. For instance, if you had two air compressors with 25gal tanks, 150psi cutoff pressures, and both with a 2HP motor: If due to differences in the compressors and construction of the units, the first unit charges the 25gal tank to 150psi in 40 seconds, and the second unit charges in in 50 seconds, then by the formula HP=GPMxPSI/1714 the two units would have a different "horsepower" despite having the same size motor.
An interesting comment on this is at http://www.truetex.com/aircompressors.htm where there are other links.