nt.number theory - Bound on the number of prime factors of logarithmically rough numbers

The intuition is that $log_{clog N}N=frac{log N}{log(clog N)}=frac{log N}{log c+loglog N}$ which is asymptotically $frac{log N}{loglog N}$. For this to work, we need that the kth prime for $k=frac{log N}{log(clog N)}+pi(clog N)approxfrac{log N}{loglog N}$ to be 'close' to $clog N$ -- actually, any constant multiple of $log N$ will do.

$p_kapproxfrac{log N}{loglog N}logfrac{log N}{loglog N}approxlog N$, so $log_{p_k}Napproxfrac{log N}{loglog N}$, as desired. This could probably made explicit with Rosser's theorem and/or Dusart's various bounds on $p_n$ and $pi(n)$.