this Q is very easily generalizable. For example consider $1,2,3$ and $4$. We have $1=1, 2=2, 3=3, 4=4, 5=4+1, 6=2\times 3, 11=4\times 3-1$. You can only use each number once. 37 is the first number not achievable. Does 1,2,3,4 form an optimal set?
EDIT: Optimal: Assuming I'm right about 37, is there a ($n=4$) set which lasts for longer, e.g. 1,3,4,7 works till 40?