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A304624
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For any positive number n with binary expansion (1, b_0, ..., b_i), a(n) is the least k > 0 such that T^j(k) has the same parity as b_j for j = 0..i (where T^i denotes the i-th iterate of the Collatz function A014682).
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1
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1, 2, 1, 4, 2, 1, 3, 8, 4, 2, 6, 5, 1, 3, 7, 16, 8, 4, 12, 10, 2, 6, 14, 5, 13, 1, 9, 3, 11, 7, 15, 32, 16, 8, 24, 20, 4, 12, 28, 10, 26, 2, 18, 6, 22, 14, 30, 21, 5, 13, 29, 17, 1, 25, 9, 3, 19, 11, 27, 23, 7, 15, 31, 64, 32, 16, 48, 40, 8, 24, 56, 20, 52, 4
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OFFSET
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1,2
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COMMENTS
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For any n > 0, a(n) is the least number whose Collatz compressed trajectory starts with a succession of tripling and halvings steps encoded in the binary representation of n (beyond the leading one).
Each term appears infinitely many times as for any n > 0 either a(2*n) or a(2*n + 1) equals a(n).
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LINKS
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FORMULA
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a(2^k) = 2^k for any k >= 0.
a(2^(k+1) - 1) = 2^k - 1 for any k >= 1.
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EXAMPLE
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The first terms, alongside the binary representation of n and some initial terms of the Collatz compressed trajectory of a(n), are:
n a(n) bin(n) traj(a(n))
-- ---- ------ ---------------------
1 1 1 (...)
2 2 10 ( 2, ...)
3 1 11 ( 1, ...)
4 4 100 ( 4, 2, ...)
5 2 101 ( 2, 1, ...)
6 1 110 ( 1, 2, ...)
7 3 111 ( 3, 5, ...)
8 8 1000 ( 8, 4, 2, ...)
9 4 1001 ( 4, 2, 1, ...)
10 2 1010 ( 2, 1, 2, ...)
11 6 1011 ( 6, 3, 5, ...)
12 5 1100 ( 5, 8, 4, ...)
13 1 1101 ( 1, 2, 1, ...)
14 3 1110 ( 3, 5, 8, ...)
15 7 1111 ( 7, 11, 17, ...)
16 16 10000 (16, 8, 4, 2, ...)
17 8 10001 ( 8, 4, 2, 1, ...)
18 4 10010 ( 4, 2, 1, 2, ...)
19 12 10011 (12, 6, 3, 5, ...)
20 10 10100 (10, 5, 8, 4, ...)
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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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AUTHOR
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STATUS
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approved
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