A335844 -id:A335844 - cp's OEIS frontend

A343356 Squares visited by a knight moving on a square-spiral with numbers equal to the ordered prime factors 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, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 5, 3, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 5, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 2, 2, 2, 2, 2, 2, 2, 2, 3, 2, 2, 2, 2, 2, 5, 2, 2

Offset: 1

Scott R. Shannon, Apr 12 2021

Many of the visited squares are numbered 2 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 369 steps the square with spiral number 3, with ordered spiral number 522, is reached after which all eight adjacent squares have been visited. The visited square with the largest spiral number is 41.
See A343385 for the visited squares given as the ordered spiral numbers.

			The square-spiral starts with 1 and is then numbered with the ordered prime factors of the positive integers as follows:
.
  11---5---2---3---3   .
   |               |   .
   2   2---2---3   2   2
   |   |       |   |   |
   2   5   1---2   2   2
   |   |           |   |
   3   2---3---7---2   2
   |                   |
  13---2---7---3---5---2
.
a(1) = 1, the starting square of the knight.
a(2) = 2. Four squares the knight can step to from the starting square are numbered 2, all of which are the same distance form the origin, so the 2 with the lowest spiral number is chosen. This is the 2 at coordinates (2,-1) relative to the starting square which has an ordered spiral number of 10.
a(35) = 3. This is the first time a square greater than 2 is stepped to. The available squares after 33 steps are 3, 3, 3, 11, 5, and 47, and the 3 at coordinates (1,4) relative to the starting square is chosen because it is the closest number to that square.
a(365) = 41. This is the largest numbered square that is stepped to. The available squares after the 363rd step are 41, 157, 313, and 43, and 41 is the smallest of these.
a(370) = 3. This is the final square stepped to as no further unvisited square is available.

N. J. A. Sloane and Brady Haran, The Trapped Knight, Numberphile video (2019).
Scott R. Shannon, Image showing the 370 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.

Cf. A343385, A343388, A316667, A323714, A323808, A329519, A329520, A335844.

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.

Original entry on oeis.org

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

Scott R. Shannon, Apr 13 2021

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.

			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.

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.

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

A343385 The ordered square spiral numbers visited by the knight in A343356.

Original entry on oeis.org

1, 10, 29, 2, 5, 40, 7, 4, 11, 26, 83, 174, 123, 84, 27, 12, 15, 18, 35, 64, 97, 36, 65, 142, 63, 66, 149, 102, 143, 146, 103, 100, 141, 62, 33, 58, 55, 28, 87, 178, 131, 92, 31, 54, 13, 60, 139, 248, 191, 316, 247, 136, 91, 182, 185, 132, 237, 300, 371, 450, 295, 228, 173, 82, 49, 78, 45, 74

Offset: 1

Scott R. Shannon, Apr 13 2021

This is the ordered square-spiral numbers visited by a knight on a square spiral as numbered in A343356. See that sequence for further details.

Scott R. Shannon, Image showing the 370 visited squares. The spiral is numbered with the ordered spiral numbers. 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.

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

A343389 The ordered square-spiral numbers visited by the knight in A343388.

Original entry on oeis.org

1, 24, 11, 4, 9, 6, 15, 2, 7, 46, 21, 72, 75, 42, 19, 38, 105, 202, 151, 104, 147, 262, 199, 102, 67, 36, 61, 32, 3, 28, 51, 124, 85, 128, 53, 30, 59, 96, 141, 250, 189, 316, 251, 392, 315, 474, 563, 660, 769, 1006, 767, 658, 879, 762, 555, 462, 553, 756, 873, 998, 761, 556, 381, 460, 305

Offset: 1

Scott R. Shannon, Apr 13 2021

This is the ordered square-spiral numbers visited by a knight on a square spiral as numbered in A343388. See that sequence for further details.

Scott R. Shannon, Image showing the 359 visited squares. The spiral is numbered with the ordered spiral numbers. 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.

Cf. A343388, A343356, A316667, A323714, A323808, A329519, A329520, A335844.

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A343385 The ordered square spiral numbers visited by the knight in A343356.

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A343389 The ordered square-spiral numbers visited by the knight in A343388.

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