Summing elements of a sequence Let the sequence $a_n$ be defined as $a_n = 2^n$ where $n = 0, 1, 2, \ldots $
That is, the sequence is $1, 2, 4, 8, \ldots$
Now assume I am told that a certain number is obtained by taking some of the numbers in the above sequence and adding them together (e.g. $a_4 + a_{19} + a_5$),  I may be able to work out what numbers from the sequence were used to arrive at the number.  For example, if the number given is 8, I know that the sum was simply $a_3$, since this is the only way to get 8.  If i'm told the number is 12, I know that the sum was simply $a_3 + a_2$. 
If I'm told what the resulting number, can you prove that I can always determine which elements of the above sequence were used in the sum to arrive at that number?  

A result which is (or should be)
well-known and which I proved many years ago as an undergraduate is this:
Suppose $(b_i)_{i=0}^{\infty}$ is an strictly increasing
series of integers with $b_0 = 1$ called the bases.
Usually $b_i = b^i$ where $b \ge 2$ is an integer.
Then every positive integer $n$ can be represented
in the form $\sum_{i=0}^{M(n)} a_i b_i$
where $M(n)+1$ is the number of "digits"
in the representation of $n$
in the "base" $(b_i)_{i=0}^{\infty}$,
the $a_i$ are integers and 
$0 \le a_i < b_{i+1}/b_i$;
the representation is unique for all integers if and only if
$b_{i+1}/b_i$ is an integer for all $i$.
This shows why the usual binary, ternary, ..., decimal representations,
where $b_i = b^i$ for an integer $b$,
are unique. It also shows that the "factorial" representation,
where $b_i = (i+1)!$ is unique.
I did this as an answer instead of a comment because
comments are too limiting.
