I see I have no takers. This is the approach that I had in mind. We want to choose as many numbers as possible from 1 to 100 without any of them being twice as large as any other.

Start by choosing 100. We can also choose 99 and 98 and, hold on. We can choose all the numbers from 51 to 100. Obviously none is twice as large as any other.

Continuing, we can’t choose 50, since 100 is twice 50, we can’t choose 49, and we can’t choose any number down to 26.

So we have – include 51 to 100, exclude 26 to 50. Here is the fun part. We can now choose 13 to 25, because twice these numbers go from 26 to 50, which were excluded.

Continuing – exclude 7 to 12, include 4 to 6, exclude 2 to 3, include 1.

Summing up, we get 1, 4, 5, 6, 13 to 25, and 51 to 100, giving us 4 + 13 + 50 = 67 of the numbers from 1 to 100.

The approach started by @ragingloli also works, but may be a little more difficult to undersand.. We use the fact that any number can be expressed as (2**k)x where x is odd. (See if you can convince yourself that this is true.) If we choose only even values of k, no number will be twice as large as any other. Start by including all the odd numbers, corresponding to k=0. We don’t want twice any of these numbers, but we can include 4 times the odd numbers (corresponding to k=2) giving 4x for odd values of x from 1 to 25. We can’t include 8 times the odd numbers, but we can include 16 (16×1 and 16×3 and 16×5) and 64 (64×1) times the odd numbers. We end up with:

All odd numbers

4x for odd values x from 1 to 25

16, 48 and 80

64

This gives 50 + 13 + 3 + 1 = 67 values. The same number as before, but with different values.