Is there a power of 2 if rearranged form another powers of 2. Powers of 2 with trailing zeros is not counted.

Among the natural numbers, I have no idea. Among the integers, I found two:

2^-3 = 0.125

2^9 = 512

2^-4 = 0.0625

2^8 = 256

Among the real numbers, there's infinitely many. Here's just one:

2^(log2(14)) = 14

2^(log2(41)) = 41

Can somebody check for the natural numbers? Even a little coding wouldn't hurt