<s>[INST] Calculating all possible sums of the numbers $2^0, 2^1, \ldots, 2^{(n-1)}$ Using the simple equation $2^{n-1}$ you get values such as:
$1,2,4,8,16,32,64,128,256,$ etc. 
How can I find all possible number combinations within this range? For example, the numbers $1,2,4,8$ give:
\begin{align}
1 &= 1 \\
2 &= 2 \\
1 + 2 &= 3 \\
4 &= 4 \\
1 + 4 &= 5 \\
2 + 4 &= 6 \\
1 + 2 + 4 &= 7 \\ 
8 &= 8 \\
1 + 8 &= 9 \\
2 + 8 &= 10 \\
1 + 2 + 8 &= 11 \\
4 + 8 &= 12 \\
1 + 4 + 8 &= 13 \\
2 + 4 + 8 &= 14 \\ 
1 + 2 + 4 + 8 &= 15
\end{align}
The numbers calculable are: $1,2,3,4,5,6,7,8,9,10,11,12,13,14,15$.
Background I'm a computer programmer trying to get a unique number which represents a series of other numbers without the unique IDs being lost. 
If I assign $5$ people with the id's, $1,2,4,8,16$, then in order to pass the list of them I don't need to list them all, just the mathematical equation to work it out. For example, if I wanted people $1,2,4$ then I could just pass the number $7$ and therefore it would HAVE to be people $1,2,4$ as there is no other possible combination to make this. 
Please let me know if my logic is flawed somewhere! 
Thanks,
Nick
 [/INST] Computing the decomposition of a nonnegative integer into sums of powers of $2$ is essentially determining the binary (base-$2$)-representation of a number.
For example, we can decompose $42$ as
$$42 = 32 + 8 + 2 = 1 \cdot 2^5 + 0 \cdot 2^4 + 1 \cdot 2^3 + 0 \cdot 2^2 + 1 \cdot 2^1 = 0 \cdot 2^0,$$
so in binary we write $42$ in bits (the binary analogue of digits) as
$$101010_2$$
(the subscript $_2$ here just reminds us that this should be read as a binary expression and not as a decimal one). Compare this decomposition with the decimal decomposition $$42 = 4 \cdot 10^1 + 2 \cdot 10^0 ;$$ we could thus write $42$ as $42_{10}$ but generally suppress a subscript $_{10}$ unless we're writing about multiple bases at once.
More generally, given the ($n + 1$) numbers $1, 2, 4, \ldots, 2^n$, the numbers expressible as sums of these are precisely the nonnegative integers with at most $n$ bits in their binary representations. (As usual, we take the empty sum to have value $0$.) These are precisely the nonnegative integers no larger than $${\underbrace{1 \cdots 1}_{n + 1}}{}_2 = 2^n + 2^{n - 1} + \cdots + 2 + 1 = 2^{n + 1} - 1.$$
By construction, this representation of any such number is unique (for the same reason that any given positive integer can only be written in decimal in a single way).
Then, in the encoding you describe, the binary representation $a_n \cdots a_0$ encodes the subset of $\{0, \ldots, n\}$ consisting precisely of the numbers $k$ for which $a_k = 1$. (In fact, this prescribes an easy algorithm for this subset from a given value of an $\texttt{integer}$-type variable.)
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