TITLE: The Nth number with M prime factors
QUESTION [5 upvotes]: Hi.
Suppose we arrange all natural numbers in a matrix P defined as follows:
P[I][J] = The Jth number with I prime factors. So P looks something like:
1
2 ,    3 ,    5 ,    7 ,   11 ,   13 ,   17 ,   19 ,   23 ,   29 ,   31 ,   37 ,   41 ,   43 ,   47 , ...
4 ,    6 ,    9 ,   10 ,   14 ,   15 ,   21 ,   22 ,   25 ,   26 ,   33 ,   34 ,   35 ,   38 ,   39 , ...
8 ,   12 ,   18 ,   20 ,   27 ,   28 ,   30 ,   42 ,   44 ,   45 ,   50 ,   52 ,   63 ,   66 ,   68 , ...
16 ,   24 ,   36 ,   40 ,   54 ,   56 ,   60 ,   81 ,   84 ,   88 ,   90 ,  100 ,  104 ,  126 ,  132 , ...
32 ,   48 ,   72 ,   80 ,  108 ,  112 ,  120 ,  162 ,  168 ,  176 ,  180 ,  200 ,  208 ,  243 ,  252 , ...
64 ,   96 ,  144 ,  160 ,  216 ,  224 ,  240 ,  324 ,  336 ,  352 ,  360 ,  400 ,  416 ,  486 ,  504 , ...
I noticed that P[i][j] = P[i-1][j]*2 if and only if j < O(1.666^i).
Examples:
i =  2 AND j <   2
i =  3 AND j <   4
i =  4 AND j <   7
i =  5 AND j <  13
i =  6 AND j <  22
i =  7 AND j <  38
i =  8 AND j <  63
i =  9 AND j < 102
i = 10 AND j < 168
i = 11 AND j < 268
i = 12 AND j < 426
I suppose that there is a more accurate approximation of the condition above.
What work has been previously done on the relation between "The Nth number with M prime factors" and "The Nth number with M-1 prime factors"?
Thanks

REPLY [2 votes]: Each sequence of "$n$-almost-primes" opens with a string of even numbers that are twice the entries that begin the previous sequence.  The first odd $n$-almost-prime is $3^n$.  Thus the sequence the OP has observed, $2,4,7,13,22,38,63,\ldots$, is basically the sequence A078843 $(3,5,8,14,23,39,64,\ldots)$ in the OEIS.  There's a recursive formula given there, attributed to Max Alekseyev:
$$a(n) = a(n-1) + \text{appi}3(n-k,[3^n/2^k]),$$
 where $k = \text{ceil}(nc)$ with $c = \log(5/3)/\log(5/2) = 0.55749295\ldots$ and $\text{appi}3(k,n)$ is the number of $k$-almost-primes not divisible by $3$ and not exceeding $n$.  This might supply the "more accurate approximation" the OP referred to.  The ratio of consecutive terms in A078843 seems to be tending to around $1.53$-something, far less than the OP's $1.666$, but all that could be quite illusory.
The link from the OEIS entry to the MathWorld entry on "Almost Primes" may also be of interest.