A344467 Numbers that return to the origin when performing a non-backtracking walk on a 2D square lattice where at each step the walk moves as close as possible to the origin and the step lengths are the ordered prime factors of the integers, starting with a step of length 1.

4, 28, 39, 190, 794, 4656, 17064, 130800, 1753625, 5154759

Offset

1,1

Comments

Starting at the origin of a 2D square lattice, take a step of length 1 in any of the four available directions. From then on take a step of length equal to the next ordered prime factor of the integers, see A027746, in a direction that takes the walk as close as possible to the origin, but without backtracking on the previous step. The sequence lists the numbers where one or more prime factors return to the origin during the walk.

Assuming another term exists it is at least 3.3*10^9.

Diagonal jumps are forbidden: for example, after the first return-to-origin, the next step is of length 5, but the walk jumps to (5,0), or a 90-degree rotation thereof, rather than (3,4) or (4,3) or any of their 90-degree rotations. Any subsequent terms exceed 25*10^9. - Lucas A. Brown, Mar 08 2024

Links

Table of n, a(n) for n=1..10.

Lucas A. Brown, Python program.

Scott R. Shannon, Image of the path up to the factors of 100. Each step is shown as a dot, and the path between steps as a line. The colors are graduated across the spectrum from red to violet to show the relative step ordering. The origin is marked with a white dot. In this and the below images if the walk stops on an axis or the four diagonals it then steps in a clockwise direction unless forbidden by the previous step. If the walk returns to the origin then the next step is upward.

Scott R. Shannon, Image of the central 200 X 200 region of the path up to the factors of 5154759. Zooming in shows a line of violet dots crossing the origin three units apart which are the factors of 3 of 5154759. Note the vast majority of dots near the origin are red or orange, indicating it is rare for numbers over one million to approach the central region.

Scott R. Shannon, Image of the full path up to the factors of 5154759.

Example

4 is the first term. After the first step of length 1 the next steps are of length 2,3,2,2: the ordered prime factors of 2,3,4. Assume the first step is upward to (0,1) and, if a choice of directions is available, it steps in a clockwise direction unless forbidden by the previous step - this direction choice is irrelevant to the sequence numbers. Given this the next steps are to coordinates (2,1), (2,-2), (0,-2), (0,0), and the second prime factor 2 of 4 returns to the origin.
28 is a term. After 4 returns to the origin the next steps up to the factors of 28 are of length 5,2,3,7,2,2,2,3,3,2,5,...,3,2,2,7. The coordinates stepped to are then (0,5),(2,5),(2,2),(2,-5),(0,-5),...,(0,-7),(0,-4),(0,-2),(0,0),(0,7),..., and the second factor 2 of 28 returns to the origin.

Prog

(Python)_See LINKS.

Crossrefs

Cf. A027746, A332939, A343153, A309500, A000040.

Sequence in context: A043074 A137314 A032405 * A307046 A179279 A061428

Adjacent sequences: A344464 A344465 A344466 * A344468 A344469 A344470

Keyword

nonn,more,hard

Author

Scott R. Shannon, May 20 2021

Status

approved