The f-number is independent of the focal length.  Actually, to be more
correct, the calculation of the f-number has already taken into account the
focal length in relation to the diameter of the effective aperture.  The
effect is that f/5.6 is f/5.6, whether you're at 20mm or 200mm, and your
exposure settings remain the same.

 There will be some difference in the actual transmissivity of various
lenses, but the difference is, for the most part, negligible.

steve

For a given f/ number and shutter period, the same quantity of light
reaches the film / sensor regardless of focal length.

The f/number is the ratio of the "opening" diameter to the focal length.
 It is an elegant way to get us away from worrying about how much light
reaches the film for a given focal length and aperture diameter.  Since
the area of the aperture increases as a function of the square of the
diameter, then a doubling in diameter, quadruples the amount of light.
Doing the math for this based on focal lenght / diameter would be an
ongoing pain in the mental butt as different focal lengths are used, so
f/ numbers allow the photog to think in terms of 'doublings' and
'halvings' with each full stop of aperture, shutter speed or ISO
setting. (to put it very simply).  This also allows fast 'trade space'
(reciprocity) decisions when considering depth of field and shutter
speed tradeoffs.

The only caveats to this are:

i: that some lens formulations are more or less transmissive.  But the
difference is not signifcant in the great majority of lenses.

ii: a very few lenses also have "transmission stops" indicationg how
much light reaches the film.  With such lenses (the Minolta 135 STF
f/2.8 [T4.5] being a good example), the depth of field is controlled by
the aperture, and the exposure is controlled by the T-stop.  (The T-stop
is always at least as slow as the aperture.  For the cited lens, the DOF
could be as shallow as f/2.8 but could never be exposed faster then
f/4.5)  [The benefit of this lens is extraordinarly smooth out-of-focus
backgrounds and tack sharp in-focus subject].

iii: in digital sensors, from camera model to camera model, there is
significant (2/3 of a stop range, perhaps more) difference in actual
sensitivity according to ISO number.  IOW one camera's ISO 100 might be
as sensitive as another camera's ISO 160.

Pedantic note: the actual calulation used by the manufacturer is
"effective" diameter as for most lenses the aperture is not round but a
polygon defined by the straight aperture blades.  Some lenses use curved
blades so that through a range of stops the aperture is nearly circular.

Cheers,
Alan

   While it is a slight simplification, you can say that the f stop is a
measure of how much light (per unit of time) comes though the lens.  In
almost all photographic situations this will work fine.

   Adding a filter or a lens with a lot more glass and glass surfaces will
reduce the total amount of light slightly and the f stop does not account
for this at all.  The f stop assumes perfect transmission with no losses.
Usually the losses for these factors are very low and too low to measure
under normal photographic work.

Just to add a little confusion to what the others have said, the f number
printed on a lens is not necessarily accurate. Some zooms, although they may
have only a single f number printed on them, may actually vary somewhat
throughout their range by as much as a stop.