For example, given four distinct numbers $5, 6, 7, 9$ We have $2^4-1$ sums, $5, 6, 7, 9, 11, 12, 14, 13, 15, 16, 18, 22, 20, 21, 27$ which are the sums from the numbers.

But if given five distinct numbers $5, 6, 7, 8, 9$, then we see $5 + 9 = 6 + 8$, hence the possible distinct sums of them cannot be easily counted by $2^5-1$.

Is it possible to easily count all the possible distinct sums from distinct numbers? Thanks!

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  • $\begingroup$ $2^4$ and $2^5$ if you count $0$ as a sum too. $\endgroup$ – Henno Brandsma Sep 30 '18 at 6:06

It's rather easy to count the total number of possible sums this way. But the actually different sums is a different matter: the knapsack problem says it's NP hard to determine whether some number $n$ is among those sums for a concrete set of numbers.

For superincreasing sequences all sums will be different. This has been the idea of building a public key encryption system, in fact: to modify a superincreasing sequence so that we still keep the unique sums property (so we have the maximal number of sums). $1,2,4,8, \ldots$ is superincreasing.

Counting the number of different sums in other cases could turn out to be pretty hard, I don't expect a closed formula for this to exist. But an algorithm does exist, see here e.g.

Also see this question and its answers

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