Calculate Row number from number grid I have the following number grid, which I believe has a cell for every number in $\Bbb N$ (not proven).
The way this grid is built is simple.
The first row holds every odd number in $\Bbb N$
The other rows are the double the value of the number the row above.
Since every number exept for prime numbers is dividable with an integer, and all prime numbers exept for 2 are present in the first row all other numbers must be present as well.
So my first question is, if I'm right assuming, that every number in this grid is unique, no number is 2 or more times in this grid.

+---++----+----+----+-----+-----+-----+-----+
|1. || 1  | 3  | 5  | 7   | 9   | 11  | ... |-> +2
+---++----+----+----+-----+-----+-----+-----+
|2. || 2  | 6  | 10 | 14  | 18  | 22  | ... |-> +4
+---++----+----+----+-----+-----+-----+-----+
|3. || 4  | 12 | 20 | 28  | 36  | 44  | ... |-> +8
+---++----+----+----+-----+-----+-----+-----+
|4. || 8  | 24 | 40 | 56  | 72  | 88  | ... |-> +16
+---++----+----+----+-----+-----+-----+-----+
|5. || 16 | 48 | 80 | 112 | 144 | 176 | ... |-> +32
+---++----+----+----+-----+-----+-----+-----+
| : || :  | :  | :  | :   | :   | :   | :   |
+---++----+----+----+-----+-----+-----+-----+
       |    |    |    |     |     |
       V    V    V    V     V     V
      *2   *2   *2   *2    *2    *2

My second question depends on the correctness of my assumption. Therefore every number in the grid needs to be unique.
I noticed that this grid is special regarding the first value of any row and the value I needed to add to get to the next number.
all numbers in row 1 can be described as $2k-1=x$, all numbers in row 2 as $4k-2=x$ , all numbers in row 3 as $8k-4=x$ and so on, so basically as $$2^rk+2^{r-1}=x,\; r \in \Bbb N,\; k \in \Bbb N,\; x \in \Bbb N$$ where $r$ represents the row number.
My 2nd question is, how I would calculate the row number with only $x$ as input number given.
I tried using wolfram alpha, but the result isn't enough since I (or we) don't know $k$.
Any help is really appreciated :-)

The answer to both your questions is based on the observation that every number can be written uniquely as $x = 2^rk$ for $k$ odd,  $r \geq 1$.
So $x$ will only ever appear in the $r+1$ th row and the column with $k$ at its top, i.e. the $(k+1)/2$ th column.
