Here's another way to rearrange the integers.

Every nonnegative integer can be written uniquely as the sum of distinct
powers of two. This is the basis for the binary representation of numbers.

Every positive integer can be written uniquely as the product of distinct
numbers of the form p^2^k where p is prime and k is a positive integer.

So consider a binary representation based on products.

Have places:

 ...  11  9   7   5   4   3   2
________________________________
     |  |   |   |   |   |   |  |
     |  |   |   |   |   |   |  |
-----+--+---+---+---+---+---+--+

with ones if the number is included in the product and zeroes if not.

Thus 1 would be written as 0
     2 as 1
     3 as 10
     4 as 100
     5 as 1000
     6 as 11
     ...

Now take these and interpret them as standard binary and you get
0, 1, 2, 4, 8, 3, 16, 5, 32, 9, ...
(A052331)

See also A052330 for the inverse and A050376 for the list of p^2^k's.

Of course this is not quite a proper rearrangement since the domain is
the positive integers and the range is nonnegative, but it's in the
same spirit.

Best regards,
Christian