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 , ...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 , ...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 , ...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
