# Arithmetic Base Conversion

Consider a number Q in a made up base system:

The base system is as follows:

It encodes a number as a sum of odd numbers:

1 3 5 7 9 ...

If the number can be expressed as a sum of unique odds. For example, the number 16 is expressed as:

1110 = 7 + 5 + 3 + 1

The system is also redundant as 16 can also be expressed as:

11000

My question:

Given a natural number u, how can u be expressed in this system quickly if u can be expressed in the system, quickly.

• If x is an even number (except 2), write 1 + (x-1). If x is an odd number (except 1, 3, 5, 7), write 1 + 3 + (x-4). Or, if you like, x is already a sum of one odd number (x). – Billy Jul 27 '13 at 2:07

If $u$ is odd, write $u$ as the following sum of odd numbers: $$u$$ If $u$ is even (and $\geq 2$), write $u$ as the following sum of odd numbers: $$(u-1)+1$$ (Converting into a string of ones and zeros is straightforward.)
If $u$ is even and greater than two: represent $u$ as $110\dots0$, where the number of $0$s is equal to $\frac{u}{2} - 2$.
If $u$ is odd: represent $u$ as $10\dots0$, where the number of $0$s is equal to $\frac{u - 1}{2}$.
$2$ cannot be represented in this system, but is the only nonnegative number for which this is the case.