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A361144
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Lexicographically earliest sequence of positive integers such that the sums Sum_{i = 1+k*2^e..(k+1)*2^e} a(i) with k, e >= 0 are all distinct.
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7
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1, 2, 4, 5, 6, 7, 8, 10, 11, 14, 15, 17, 16, 19, 20, 21, 22, 23, 24, 26, 27, 28, 29, 30, 33, 34, 36, 37, 38, 39, 40, 42, 44, 46, 47, 49, 48, 51, 52, 53, 54, 56, 58, 60, 61, 62, 63, 64, 65, 66, 68, 69, 70, 71, 72, 74, 75, 78, 79, 81, 80, 83, 84, 85, 86, 87, 88
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OFFSET
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1,2
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COMMENTS
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In other words, a(1), a(2), a(1)+a(2), a(3), a(4), a(3)+a(4), a(1)+a(2)+a(3)+a(4), a(5), a(6), a(5)+a(6), etc. are all distinct (see A361227 for these values).
In particular, all terms are distinct (but not necessarily in increasing order).
We can arrange the terms of the sequence as the leaves of a perfect infinite binary tree, the sums with e > 0 corresponding to parent nodes; each node will contain a different value and all values will appear in the tree (if n = 2^m+1 for some m > 0, then a(n) will equal the least missing value so far in the tree).
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LINKS
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FORMULA
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Empirically, a(n) ~ 4*n/3 as n tends to infinity.
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EXAMPLE
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The first terms (at the bottom of the tree) alongside the corresponding sums are:
176
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43 133
----------------- -----------------
12 31 57 76
--------- --------- --------- ---------
3 9 13 18 25 32 35 41
----- ----- ----- ----- ----- ----- ----- -----
1 2 4 5 6 7 8 10 11 14 15 17 16 19 20 21
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PROG
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(PARI) See Links section.
(C++) See Links section.
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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