Here is an example using the tag system. Write the number in unary, then remove the first two digits and add the result of applying the rules to the first digit of the previous step; the final result is 1:

For the abstract machine, notice that an even number in base two ends in zero, like the case of 12, whose digits in binary are {1,1,0,0}. Remove the 0s to get 3 with digits {1,1}, corresponding to halving 12 and 6, then add {1,1} and {1,1,1} (a 1 is appended at the end), equivalent to the operation

, yielding 10 with digits {1,0,1,0}. Then repeat the operation.

For the correspondence with rational numbers, represent the integer

by

, where

is the highest power of 2 less than

, then repeatedly apply the function

where

is the denominator and

the denominator, until the denominator is 0.

The Syracuse function works with odd integers; start with

and compute

, factor it as the product of a power of 2 and an odd integer

, which is the new term of the sequence; for example

.

For more details about the implementation of most of the formulations presented, see the Related Links and [1].