Numbers with a prime factor > sqrt

[dodo]

Suppose you divide all non-prime numbers in two categories, those which (a) have a prime factor greater than the square root of the number, and those which (b) don't, and all prime factors are less or equal than the square root.

Let Ca and Cb be the count of numbers in categories (a) and (b), resp. As you collect more numbers, a quick&dirty survey seems to indicate that the ratio Ca/Cb keeps growing (I don't know if converging), from 1.4 to 1.9 to 2.3... (Funny, actually I kind of imagined Cb to be bigger than Ca.)

What kind of math knowledge in number theory (or not) applies to the study of this? Any pointer, please?

 

[CRGreathouse]

Dickman's rho function is an asymptotic estimate of the fraction of smooth numbers. In particular, $\rho(2)$ is the asymptotic proportion of numbers that are $\sqrt n$-smooth. $\rho(2)\approx30.69\%$ so your ratio should converge.

(The rho function, unlike your function, includes primes -- but they're asymptotically negligible.)