Let $\pi_k(2^n)$ be the number of almost-primes (numbers with k factors including repetitions) less than $2^n$. I noticed that for large values of n and values of k near n, a sequence $\{\pi_k\}$ emerges. For example, for n = 17, for k = 17,...,12, the sequence is {1,2,7,15,37,84}. The terms of the sequence emerge as n grows.
The sequence is in OEIS as A052130, and there is a brief comment there that may explain the sequence. Could someone elaborate a bit on the comment or provide something a little more substantive?
Thanks.