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 , ...][1]

   [8 ,   12 ,   18 ,   20 ,   27 ,   28 ,   30 ,   42 ,   44 ,   45 ,   50 ,   52 ,   63 ,   66 ,   68 , ...][2]

  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 , ...][3]

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


  [1]: http://oeis.org/A007774
  [2]: http://oeis.org/A014612
  [3]: http://oeis.org/A046306