Set of positive integers with unique sums

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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.




combinatorics elementary-number-theory




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  • 3




    try the powers of 2
    – Will Jagy
    May 6 '15 at 1:55






  • 2




    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 at 10:09















up vote
4
down vote

favorite
2












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.




combinatorics elementary-number-theory




share|cite|improve this question
















  • 3




    try the powers of 2
    – Will Jagy
    May 6 '15 at 1:55






  • 2




    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 at 10:09













up vote
4
down vote

favorite
2









up vote
4
down vote

favorite
2






2





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.




combinatorics elementary-number-theory




share|cite|improve this question












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.





combinatorics elementary-number-theory





share|cite|improve this question











share|cite|improve this question




share|cite|improve this question










asked May 6 '15 at 1:39









William

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  • 3




    try the powers of 2
    – Will Jagy
    May 6 '15 at 1:55






  • 2




    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 at 10:09













  • 3




    try the powers of 2
    – Will Jagy
    May 6 '15 at 1:55






  • 2




    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 at 10:09








3




3




try the powers of 2
– Will Jagy
May 6 '15 at 1:55




try the powers of 2
– Will Jagy
May 6 '15 at 1:55




2




2




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




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




@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 at 10:09





it looks like a general case of sumfree sets problem.
– Abr001am
Apr 17 at 10:09











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They are called sets with distinct subset sums.






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    They are called sets with distinct subset sums.






    share|cite|improve this answer
























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      They are called sets with distinct subset sums.






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        up vote
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        They are called sets with distinct subset sums.






        share|cite|improve this answer












        They are called sets with distinct subset sums.







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        answered Aug 14 at 4:05









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