I need a set that enables me to identify specific containing elements by any summation of any of its subsets (see example to understand) My question is more practically understood by example. I need a set A that behaves like the one below:
Set A: {1,3,5}
Set B (all subsets of A): {1}, {3}, {5}, {1,3}, {1,5}, {3,5}, {1,3,5}
Set C (summations of each element of B): {1}, {3}, {5}, {4}, {6}, {8}, {9}
Because every element of C is of unique value (that's the criteria), what I have is: every element of C identifies a specific combination of elements of A. If I am given 4, I know it's the sum of the specific elements 1 and 3 and no other possible combination of elements, and so forth for any given element of C.
To create a set A of 3 elements is easy, but I need the largest possible set A that contains only 2 digit natural numbers (from 01 to 99) matching this criteria. I need also to be able to find the matching elements of A from any given element of C (if I'm given 9 I need to know it refers to the elements 1+3+5 and so forth for any given sum).
Can anyone help me?

See OEIS sequence A096858.
The 8-element set {40,60,71,77,80,82,83,84} has all its subset sums distinct, so this beats Alex's 7-element answer.
