A389685 - OEIS

%I #64 Jan 27 2026 22:12:53

%S 1,2,47,4,21,94,2763,8,11049,42,1383,188,173,5526,5427,16,689,22098,

%T 22143,84,69,2766,2715,376,88569,346,1565469640171323044119,11052,

%U 11069,10854,195683705021415380515,32,354273,1378,1359,44196,44213,44286,11337451,168

%N a(n) is the unique nonnegative integer whose binary expansion is the parity sequence of the Collatz orbit of n, interpreted through a particular conjugacy (see Comments).

%C This is the sequence xi(H(n)) shown on p. 10 of Monks (2025).

%C Let b(k) be the parity of the k-th Collatz iteration on n. Then, a(n) = Sum_{k>=0} c(k)*2^k, where the coefficients c(k) are uniquely determined by requiring the polynomial H(n) = Sum_{k>=0} c(k)*q^k over F_2[q] have parity vector b under the map T_{1,1+q^2}(x) = x/q if x is even and (x+1+q^2)/q if x is odd.

%H Sean A. Irvine, <a href="https://github.com/archmageirvine/joeis/blob/master/src/irvine/oeis/a389/A389685.java">Java program</a> (github)

%H Kenneth G. Monks, <a href="https://arxiv.org/abs/2508.10153">On q-analogs of the 3x+1 Dynamical System</a>, arXiv:2508.10153 [math.NT], 2025. See pp. 10, 12 (Table 2).

%Y Cf. A006370.

%K nonn

%O 1,2

%A _Michael De Vlieger_, Jan 27 2026