Yes, and in small cases you can work out the distribution explicitly. For example: - 1/6 of all have their second smallest prime factor equal to 3 (that is, th

Yes, and in small cases you can work out the distribution explicitly. For example: - 1/6 of all have their second smallest prime factor equal to 3 (that is, they have a factor of 2 and a factor of 3) - 1/5 of integers have a factor of 5; of those, 1/2 have a factor of 2 or a factor of 3, but not both (that is, they're congruent to 2, 3, or 4 mod 6); so 1/10 of integers have second smallest prime factor equal to 5. So the proportion of integers with second smallest prime factor equal to 5 is 1/6 + 1/10 = 8/30. Thus the first quartile of second smallest primes is 5, and similarly for any quantile between 1/6 and 8/30. Don't ask me to work out the third quartile - this distribution has a really long tail. In particular I suspect the limiting distribution has no mean.