Out of the problems on the app, it has been proven you can get up to 8 for original (check all cases), infinite for odd (see below), 16 for even (as amount for original doubled), 24 for multiples of 3 (as amount for original tripled), and infinite for non-multiples of 5 (see below), cubes (see below), prime (see below), and fibonacci (see below).
If you put all the odds on one side, then it will work because odd+odd=even.
For non-multiples of 5:
Put all the numbers ending with 1,4,6,9 on one side and all the numbers ending with 2,3,7,8 on the other side. For the last digit sums, 1+4=5, 1+6=7, 1+9=0, 4+6=0, 4+9=3, 6+9=5 so the first side is safe. 2+3=5, 2+7=9, 2+8=0, 3+7=0, 3+8=1, 7+8=5, so the second side is also safe.
For cubes and nth powers, see this:
Put them on one side except if the number is the second member of a twin prime (for 3,5,7, put 3 and 7 on one side and 5 on the other). It will work because of the odd theorem and because the gap between the numbers on one side must be greater than 2.
Put them any way without having 3 in a row on one side.
But squares remain unsolved. Please share if you have any results or questions!