A343388 Squares visited by a knight moving on a square-spiral with numbers equal to the ordered divisors of the positive integers and where the knight moves to the smallest numbered unvisited square; the minimum distance from the origin is used if the square numbers are equal; the smallest ordered spiral number is used if the distances are equal.

1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 2, 1, 1, 4, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 3, 1, 1, 1, 2, 2, 1, 1, 1, 2, 1, 5, 1, 1, 3, 1, 4, 1, 1, 8, 1, 3, 2, 2, 3, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 1, 1, 1

Offset

1,9

Comments

Many of the visited squares are numbered 1 due to the large number of such terms on the board and the knight's preference for the lowest available numbered square.

The sequence is finite. After 358 steps the square with spiral number 13, with ordered spiral number 37, is reached after which all eight adjacent squares have been visited. The visited square with the largest spiral number is 28.

See A343389 for the visited squares given as the ordered spiral numbers.

Links

Table of n, a(n) for n=1..99.

Scott R. Shannon, Image showing the 359 visited squares. The starting square is highlighted in white, the visited squares in yellow, the path is colored across the spectrum to show the relative step ordering, and the final square is highlighted in red.

Example

The square-spiral is numbered with the ordered divisors of the positive integers as follows:
.
   1---7---1---6---3   .
   |               |   .
   2   3---1---2   2   11
   |   |       |   |   |
   4   1   1---1   1   1
   |   |           |   |
   8   2---4---1---5   10
   |                   |
   1---3---9---1---2---5
.
a(1) = 1, the starting square of the knight.
a(2) = 1. One square numbered 1 can be stepped to from the starting square, the square with coordinates (1,-2) relative to that square.
a(9) = 2. This is the first time a square greater than 1 is stepped to. The available squares after 7 steps are 3, 11, 10, 2, 9, 2, and 3. The 2 at coordinates (-1,-1) relative to the starting square is because it is the closest number to that square.
a(146) = 28. This is the largest numbered square that is stepped to. The available squares after the 144th step are 117, 213, 47, 70, 61, and 28, and 28 is the smallest of these.
a(359) = 13. This is the final square stepped to as no further unvisited square is available.

Crossrefs

Cf. A343389, A343356, A316667, A323714, A323808, A329519, A329520, A335844.

Sequence in context: A318666 A301391 A068696 * A133009 A210705 A291771

Adjacent sequences: A343385 A343386 A343387 * A343389 A343390 A343391

Keyword

nonn,fini

Author

Scott R. Shannon, Apr 13 2021

Status

approved