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I require a set of numerical elements on which the sum of some of these elements is unique to the set, that it's to say, no other combination in the sum of elements will result the same outcome.

Assuming this is a common occurrence, is there a name to such sequence or phenomenon?

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They are called sets with distinct subset sums.

There are many open problems about them or related concepts. See http://www.openproblemgarden.org/op/sets_with_distinct_subset_sums or Tao and Vu's book Additive Combinatorics for some such open problems.

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