Is there a scheme by which we can form distinct sums of sums.

Question

In the same way that we can form, or alternatively pull back apart, sums of powers of two -

is there a way to form sums of sums of values, in any sort of pattern or scheme, such that we could pull them back apart into their original values?

Answer 1

If I interpret your question correctly, you consider a finite set $S = \{n_1, \ldots, n_k\}$ of nonnegative integers and you observe that if you know the sum $2^{n_1} + \dotsm + 2^{n_k}$, then you can recover each individual power $2^{n_i}$ (for $1 \leqslant i \leqslant k$) and hence the set $S$.

Note that the integers $n_i$'s have to be all distinct, otherwise the property may fail. Coming back to your question, you could take powers of any integer $a\geqslant 2$: if you know the sum $a^{n_1} + \dotsm + a^{n_k}$, you will be able to recover each individual power $a^{n_i}$ and consequently the set $S$ (think of $a = 10$ as an example).