Probably not. We are talking about some function of first N numbers and its limit when N approaches infinity. There is more than one reasonable way to clarify "

Probably not. We are talking about some function of first N numbers and its limit when N approaches infinity. There is more than one reasonable way to clarify "first prime factor" and "first N numbers", but I think for all of these reasonable definitions the median diverges, having different subsequential limits for even and odd values of N.