F/#, Working F/# and Numerical Aperture

numerical aperture = sin(theta)

Where theta is half the cone angle subtended by rays that enter or exit an optical system. The F/# is defined as the ratio between the lens aperture (D) and its focal length f.

F/# = f/D

For small theta values:

F/# = 1 /(2 *numerical aperture)

hence

numerical aperture = 1/(2 * F/#)

Note that numerical aperture (and F/#) refer to both image and object space, as they can define both the cone angle of incoming and outgoing rays. Usually F/# refers to image space and numerical aperture is more commonly used in object space (incoming rays).
In macro lenses, like Telecentric Lenses, the F/# parameter loses its meaning as the object is not located at infinity; the working F/# should be used instead. Those two parameters come together in the formula:

Working F/# = (1 + magnification) * F/#

Note also that

numerical aperture(object) = magnification * numerical aperture(image)

and consequently

working F/#(object) = working F/#(image) / magnification.