A336413 Squares visited by a chess rook moving on a square-spiral numbered board where the rook moves to the closest unvisited square containing a prime number. In case of a tie it chooses the square with the smallest prime number.

1, 2, 3, 5, 7, 41, 43, 109, 107, 103, 37, 193, 191, 97, 101, 199, 197, 683, 677, 673, 1753, 1747, 1429, 1427, 887, 883, 661, 659, 881, 877, 307, 461, 463, 653, 1129, 1733, 2083, 2081, 3323, 3319, 3797, 3793, 5419, 5417, 5413, 4297, 2861, 2857, 2447, 2069, 1723, 1721, 1409, 1123, 1117, 1399

Offset: 1

Author

Scott R. Shannon, Jul 21 2020

Keywords

• nonn
• walk
• fini
• full

Comments

This sequences gives the numbers of the squares visited by a chess rook moving on a square-spiral numbered board where the rook starts on the 1 numbered square and at each step moves to the closest unvisited square containing a prime number. The movement is restricted to the four directions a rook can move on a standard chess board, and the rook cannot move over a previously visited square. If two or more unvisited prime numbered squares exist which are the same distance from the current square then the one with the smallest prime number is chosen. Note that if the rook simply moves to the closest unvisited square the sequence will be infinite as the rook will just follow the square spiral path.

The sequence is finite. After 350 steps the square with number 2179 is visited, after which all four squares the rook can move to have been visited.

The first term where this sequence differs from A336447, where the rook steps to the smallest unvisited prime, is a(7) = 43. See the examples below.

The largest visited square is a(151) = 30539. Both the largest step distance between visited squares, 24 units, and the largest prime gap between visited squares, 6744, occur between a(229) = 2143 and a(230) = 8887. The smallest unvisited prime is 11.

Examples

			The board is numbered with the square spiral:
.
  17--16--15--14--13   .
   |               |   .
  18   5---4---3  12   29
   |   |       |   |   |
  19   6   1---2  11   28
   |   |           |   |
  20   7---8---9--10   27
   |                   |
  21--22--23--24--25--26
.
a(1) = 1, the starting square for the rook.
a(2) = 2. The four unvisited prime numbered squares around a(1) the rook can move to are numbered 2,61,19,23. Of these 2 is the closest, being 1 unit away.
a(3) = 3. The three unvisited prime numbered squares around a(2) = 2 the rook can move to are numbered 47,11,3. Both 11 and 3 are 1 units away, and of those 3 is the smallest.
a(7) = 43. The three unvisited prime numbered squares around a(6) = 41 the rook can move to are numbered 37,43,107. Both 43 and 107 are 2 units away, and of those 43 is the smallest. Note that 37, the smallest available prime, is 4 units away.
a(230) = 8887. There is only one unvisited prime numbered square around a(229) = 2143 the rook can move to. The square 8887 is 24 units away to the left of 2143.
		

Links

• Scott R. Shannon, Image showing the 350 steps of the rook's path. A green square shows the starting 1 square, a red square shows the final square with number 2179, and a thick white line is the path between visited squares. All visited prime numbered squares are shown in yellow, while those unvisited squares containing primes are shown in grey. The four squares which block the rook's movement from the final square are shown with a red border. The square spiral numbering of the board is shown as a thin white line. Click on the image to zoom in to see the prime numbers.

Crossrefs

Cf. A336402, A336446, A336447, A330979, A000040, A063826, A214664, A214665, A136626, A115258, A331027.