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A296616
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Lexicographically earliest sequence of distinct positive terms such that, for any n > 0, the binary expansion of a(n) * a(n + 1) starts with the binary expansion of n.
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1
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1, 2, 4, 3, 6, 7, 14, 8, 16, 9, 18, 5, 10, 11, 21, 12, 22, 13, 23, 27, 24, 28, 26, 29, 53, 31, 54, 32, 56, 17, 57, 35, 15, 36, 61, 37, 63, 19, 64, 39, 33, 20, 34, 41, 69, 42, 71, 43, 72, 44, 73, 45, 74, 46, 38, 47, 77, 48, 78, 49, 79, 25, 40, 51, 81, 52, 82
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OFFSET
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1,2
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COMMENTS
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It is likely that this sequence is a permutation of the natural numbers.
The lines visible in the scatterplot of the first terms seems to corresponds to set of indices n where the function f(n) = Sum_{k=1..n-1} (-1)^k * (A029837(1+a(k)*a(k+1)) - A029837(1+k)) has the same value; those lines can be partitioned into two groups, depending on the parity of n (see Links section).
This sequence has connections with A272679: here the binary expansion of a(n)*a(n+1) starts with that of n, there the binary expansion of a(n)^2 starts with that of n.
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LINKS
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EXAMPLE
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The first terms, alongside the binary representations of n and a(n) * a(n + 1), are:
n a(n) bin(n) bin(a(n)*a(n+1))
-- ---- ------ ----------------
1 1 1 10
2 2 10 1000
3 4 11 1100
4 3 100 10010
5 6 101 101010
6 7 110 1100010
7 14 111 1110000
8 8 1000 10000000
9 16 1001 10010000
10 9 1010 10100010
11 18 1011 1011010
12 5 1100 110010
13 10 1101 1101110
14 11 1110 11100111
15 21 1111 11111100
16 12 10000 100001000
17 22 10001 100011110
18 13 10010 100101011
19 23 10011 1001101101
20 27 10100 1010001000
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PROG
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(C++) See Links section.
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CROSSREFS
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KEYWORD
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nonn,base
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AUTHOR
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STATUS
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approved
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