A332939 -id:A332939 - cp's OEIS frontend

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

Scott R. Shannon, May 20 2021

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

			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.

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.

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

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A343483 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.

Original entry on oeis.org

243, 256, 256, 344, 426, 516, 876, 3886, 30420, 58852

Offset: 1

Scott R. Shannon, Apr 16 2021

This sequence uses the same walk rules as in A344467 but the first step of length 1 is removed, thus the first step is of length 2, the prime factor of 2. See that sequence for further details of the walk's rules.

Assuming another term exists it is at least 2x10^9.

			243 is the first term. It takes a surprisingly larger number of steps, 629, before the walk returns to the origin for the first time. 243 is 3^5 and after the factors of 242 the walk is at coordinate (3,-6) relative to the origin, which can be reached with three steps of length 3, the first three factors of 243.
256 is both the second and third term. The second return to the origin is forty steps after the first visit by the factors of 243, and the third is only four steps after the second - this is due to 256 being 2^8. So after the second return to the origin the walk can go back to it immediately with four steps of length 2, walking out the sides of a square.

Scott R. Shannon, Image of the path up to factors of 243. 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.

Cf. A344467, A027746, A332939, A343153, A309500, A000040.

cp's OEIS Frontend

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.

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