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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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%I #14 May 19 2018 18:53:12

%S 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,

%T 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,

%U 19,11,27,23,7,15,31,64,32,16,48,40,8,24,56,20,52,4

%N 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).

%C 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).

%C Each term appears infinitely many times as for any n > 0 either a(2*n) or a(2*n + 1) equals a(n).

%H Rémy Sigrist, <a href="/A304624/b304624.txt">Table of n, a(n) for n = 1..16383</a>

%H Rémy Sigrist, <a href="/A304624/a304624.gp.txt">PARI program for A304624</a>

%H <a href="/index/3#3x1">Index entries for sequences related to 3x+1 (or Collatz) problem</a>

%F a(2^k) = 2^k for any k >= 0.

%F a(2^(k+1) - 1) = 2^k - 1 for any k >= 1.

%F a(A081254(n)) = 1 for any n >= 1.

%e The first terms, alongside the binary representation of n and some initial terms of the Collatz compressed trajectory of a(n), are:

%e n a(n) bin(n) traj(a(n))

%e -- ---- ------ ---------------------

%e 1 1 1 (...)

%e 2 2 10 ( 2, ...)

%e 3 1 11 ( 1, ...)

%e 4 4 100 ( 4, 2, ...)

%e 5 2 101 ( 2, 1, ...)

%e 6 1 110 ( 1, 2, ...)

%e 7 3 111 ( 3, 5, ...)

%e 8 8 1000 ( 8, 4, 2, ...)

%e 9 4 1001 ( 4, 2, 1, ...)

%e 10 2 1010 ( 2, 1, 2, ...)

%e 11 6 1011 ( 6, 3, 5, ...)

%e 12 5 1100 ( 5, 8, 4, ...)

%e 13 1 1101 ( 1, 2, 1, ...)

%e 14 3 1110 ( 3, 5, 8, ...)

%e 15 7 1111 ( 7, 11, 17, ...)

%e 16 16 10000 (16, 8, 4, 2, ...)

%e 17 8 10001 ( 8, 4, 2, 1, ...)

%e 18 4 10010 ( 4, 2, 1, 2, ...)

%e 19 12 10011 (12, 6, 3, 5, ...)

%e 20 10 10100 (10, 5, 8, 4, ...)

%o (PARI) See Links section.

%Y Cf. A014682, A081254.

%K nonn,base,look

%O 1,2

%A _Rémy Sigrist_, May 15 2018