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A308093
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For any n > 0, let f_n be the lexicographically earliest sequence of distinct positive terms such that the concatenation of the binary representation of its terms, without leading zeros, corresponds to the binary representation of n repeated indefinitely. Apparently, n always appears in f_n. a(n) gives the index of n in f_n.
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1
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1, 1, 2, 1, 3, 3, 3, 1, 3, 2, 8, 3, 8, 5, 4, 1, 3, 3, 8, 3, 5, 8, 15, 3, 8, 8, 15, 5, 10, 7, 5, 1, 3, 3, 8, 2, 7, 7, 15, 3, 7, 3, 9, 7, 7, 15, 28, 3, 8, 7, 15, 7, 15, 7, 24, 5, 10, 9, 17, 7, 11, 9, 6, 1, 3, 3, 8, 3, 7, 7, 15, 3, 5, 8, 14, 8, 14, 11, 28, 3, 7
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OFFSET
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1,3
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COMMENTS
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In other words, f_n(a(n)) = n.
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LINKS
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FORMULA
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a(2^k) = 1 for any k >= 0.
a(2^k-1) = k for any k > 0.
a(n) = 2 iff n = A007582(k) for some k > 0.
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EXAMPLE
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The first terms, alongside the binary representations of n and the first terms of f_n, are:
n a(n) bin(n) bin(f_n)
-- ---- ------ -----------------------------------------
1 1 1 1,...
2 1 10 10,...
3 2 11 1,11,...
4 1 100 100,...
5 3 101 10,1,101,...
6 3 110 1,10,110,...
7 3 111 1,11,111,...
8 1 1000 1000,...
9 3 1001 100,1,1001,...
10 2 1010 10,1010,...
11 8 1011 10,1,110,11,101,1101,11011,1011,...
12 3 1100 1,100,1100,...
13 8 1101 1,10,11,101,110,1110,11101,1101,...
14 5 1110 1,110,11,10,1110,...
15 4 1111 1,11,111,1111,...
16 1 10000 10000,...
17 3 10001 1000,1,10001,...
18 3 10010 100,10,10010,...
19 8 10011 100,1,1100,11,1001,11001,110011,10011,...
20 3 10100 10,100,10100,...
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PROG
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(PARI) 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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