Without using the prime number theorem, we obtain the asymptotics of the $r$th largest prime divisor of a harmonically distributed random positive integer $N$; harmonic asymptotics are obtained from asymptotics of the zeta distribution via Tauberian methods. (Knuth and Trabb-Pardo need a strong form of the prime number theorem to obtain the distributions when $N$ is uniformly distributed.) A trick brings in Poisson variates, and then we can use the methods developed for the fractional length of the $r$th longest cycle in a random permutation.
Stuart P. Lloyd. "Ordered Prime Divisors of a Random Integer." Ann. Probab. 12 (4) 1205 - 1212, November, 1984. https://doi.org/10.1214/aop/1176993149