# Set of positive integers with unique sums

What I'm looking for is the name of a type of number set. Given a number T (for total) and a set of positive integers S, I want to uniquely identify the subset of S that sums to T. All sets containing 1 or 2 positive integers will pass the test, since there is a unique combination of those numbers and the sum will therefore be the same.

For example:

{1, 7, 89} passes the test. Any combination of those numbers, when summed, will generate a unique T, and vice versa, any T that is a sum of a combination of those numbers will generate a unique subset of S. So, the set of all T s for the above set is {1, 7, 8, 89, 90, 96}.

{2, 3, 7, 8} does not pass the test. There are multiple combinations that yield a total of 10 ({2, 8}, and {3, 7}). So if I specified a T of 10, you could not tell me with confidence the combination that produced that sum.

With that out of the way. My question is this... Is there name for a set of positive integers like this? I'd like to learn more about them more a personal project of mine.

• try the powers of 2 – Will Jagy May 6 '15 at 1:55
• What do you call a set whose subsets all have unique sums? is a similar question. Greg Martin's comment mentions Sidon sets, which seems like what you're looking for, since your examples only require unique sums for each two integers. – kate May 10 '15 at 19:54
• @kate I'd say that post answers my question pretty well! I don't want to limit myself to summing two integers (in the case of Sidon sets), but if you post an answer with a link to that post, I'll confirm it for you. =) – William May 20 '15 at 5:21
• it looks like a general case of sumfree sets problem. – Abr001am Apr 17 '18 at 10:09