A389685 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).

1, 2, 47, 4, 21, 94, 2763, 8, 11049, 42, 1383, 188, 173, 5526, 5427, 16, 689, 22098, 22143, 84, 69, 2766, 2715, 376, 88569, 346, 1565469640171323044119, 11052, 11069, 10854, 195683705021415380515, 32, 354273, 1378, 1359, 44196, 44213, 44286, 11337451, 168

Offset

1,2

Comments

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

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.

Links

Table of n, a(n) for n=1..40.

Sean A. Irvine, Java program (github)

Kenneth G. Monks, On q-analogs of the 3x+1 Dynamical System, arXiv:2508.10153 [math.NT], 2025. See pp. 10, 12 (Table 2).

Crossrefs

Cf. A006370.

Sequence in context: A196197 A273380 A124690 * A195877 A141579 A107205

Adjacent sequences: A389682 A389683 A389684 * A389686 A389687 A389688

Keyword

nonn

Author

Michael De Vlieger, Jan 27 2026

Status

approved