Hmm I would do it this way

Map 1 to 3.

Define a bijection between (1,2] and using the usual trick of specifying a point at (2,1) where the number in (1,2] represents an angle in (-pi/2, 0] from the point to the x-axis. (-pi/2 is directly to the left, 0 is straight down).

For non-integers greater than 2, map them to themselves.

For integers n greater than two, map them to n+1.