$ S = \{3,5,6,8,9,11,12,13,\cdots \}$ $ T = \mathbb Z^ + - S$ $ T = \{1,2,4,7,10\}$ if $ a \in S$ then either $ a \in \{3,5,6,8,9,11,12\}$ a sum of two different members not in $ S$ or $ a = 13 = 5 + 8$ both in $ S$ or $ a \ge 13$ then is a sum of two distinct members of $ S$ by induction because $ \{3\} \in S$ and $ \{11,12,13\} \in S$ if two distinct members are not in $ S$ then all their sums are in $ S$ $ 1 + 2 = 3,1 + 4 = 5,1 + 7 = 8,1 + 10 = 11$ $ 2 + 4 = 6,2 + 7 = 9,2 + 10 = 12$ $ 4 + 7 = 11,4 + 10 = 14$ $ 7 + 10 = 17$ If two distinct members $ a,b$ are in $ S$ then if $ a + b \ge 11 \Rightarrow a + b \in S$ if $ a + b \le 10 \Rightarrow$ WLOG $ (a < b) (a,b) \in \{(3,5),(3,6)\}$ and its sum is in $ S$ Generalization: Is there a set $ S$ of positive integers such that a number is in $ S$ if and only if it is the sum of $ k (k \ge 2)$ distinct members of $ S$ or a sum of $ k$ distinct positive integers not in $ S$?