A number $n$ with prime factorization $$n=\prod_{i=1}^rp_i^{a_i}$$
is a k-almost prime if it has a sum of exponents $$\sum_{i=1}^{r}a_i=k$$ i.e., when the prime factor (multiprimality) function $\Omega(n)=k$.
A definition for an almost prime counting function (counted with multiplicity, as opposed to distinct prime factors) may then be given as: $$ N_k(x):=\ \mid\{n\leq x : \Omega(n)=k\}\mid \ $$ If $\Omega(x)=k$ and $2x<3^{k+1}$, a limit is reached whereby $N_k(x)=N_{k+n}(2^n x)$ for $n>0$ (where $n\in \mathbb{Z}$).
How would I go about proving this / finding a proof for this? I have looked at Nicolas' paper from 1984 on the subject, but (a) it doesn't deal with this exact problem, and (b) my French is not too good! I can find no other references to proofs pertaining to the limits reached by almost primes - only passing comments on oeis.