A333713 -id:A333713 - cp's OEIS frontend

A333714 Squares visited by a chess king moving on a square-spiral numbered board where the king moves to the adjacent unvisited square containing the spiral number with the most divisors. In case of a tie it chooses the square with the highest spiral number.

1, 8, 24, 48, 80, 120, 168, 224, 288, 360, 440, 528, 624, 728, 840, 960, 1088, 1224, 1368, 1520, 1680, 1848, 2024, 2208, 2400, 2600, 2808, 3024, 3248, 3480, 3720, 3968, 4224, 4488, 4760, 5040, 5328, 5624, 5928, 6240, 6560, 6888, 7224, 7568, 7920, 8280, 8648, 9024, 9408, 9800, 10200, 10608

Offset: 1

Scott R. Shannon, Jul 02 2020

This sequence gives the numbers of the squares visited by a chess king moving on a square-spiral numbered board where the king starts on the 1 numbered square and at each step moves to an adjacent unvisited square, out of the eight adjacent neighboring squares, which contains the number with the most divisors. If two or more adjacent squares exist with the same highest number of divisors then the square with the highest spiral number is chosen. Given both of these rules tend to force the king to squares with larger numbers, and thus move away from the central 1 starting square, it is remarkable that the king is eventually trapped. Note that if the king simply moves to the highest available number the sequence will be infinite as the king will step along the southeast diagonal from square 1 forever.
The sequence is finite. After 1113 steps the square with number 855481 is visited, after which all adjacent neighboring squares have been visited.
Due to the king's preference for squares with the most divisors it will avoid prime numbers unless no other choice exists. Of the 1113 visited squares only once does it visit a square with a prime number, at a(308) = 108223. This is due to a(307) = 106913 having square 108223 as its sole neighboring unvisited square. This is the only time in the sequence where only one unvisited adjacent neighbor is available.
As even numbers >= 6 will always contain 4 or more divisors the king will tend to visit more even numbers than odd numbers; in the 1113 visited squares 929 contain an even number while only 184 contain an odd number.
As the even numbers are diagonally adjacent in the square spiral the king's path will be dominated by diagonal steps, often taking many diagonal steps in succession - see the attached link image. In fact after the first downward step to 8 the next 110 steps are along the southeast diagonal, stepping to successively larger even numbers. This sequence is finally broken on the 112th step when the square with number 50624, with 28 divisors, is the next square in the southeast direction. However the square with number 50622, with 32 divisors, is in the southwest direction so is the next square chosen. It is not until the 166th step, to the square with number 108230, that the path takes a step to a lower number than the one it is currently on.
The largest visited square is a(1050) = 942676. The visited square with the maximum number of divisors is a(680) = 388080, which has 180 divisors. The lowest unvisited square is 2.

			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 king.
a(2) = 8. The eight unvisited squares around a(1) the king can move to are numbered 2,3,4,5,6,7,8,9. Of these 6 and 8 both have the maximum four divisors, and of those 8 is the largest.
a(3) = 24. The seven unvisited squares around a(2) = 8 the king can move to are numbered 9,2,6,7,22,23,24. Of these 24 has eight divisors, the largest number.
a(113) = 50622. The seven unvisited squares around a(112) = 49728 the king can move to are numbered 50622, 49727, 50623, 48841, 50624, 49729, 48842. Of these 50622 has thirty-two divisors, the largest number. This is the step that breaks the sequence of 110 steps to the southeast direction starting from a(2) = 8.
a(308) = 108223. This is the first and only time a prime number is visited; a(307) = 106913 has square 108223 as the sole unvisited adjacent neighbor.
a(1114) = 855481. The two unvisited squares around a(1113) = 859184 the king can move to are numbered 862894 and 855481. Of these 855481 has eight divisors, the largest number. However square 855481 is surrounded by the eight squares with numbers 859183, 855480, 851785, 859184, 851786, 859185, 855482, 851787 all of which have been previously visited, so the king is trapped.

Scott R. Shannon, Table of n, a(n) for n = 1..1114
Scott R. Shannon, Image showing the 1113 steps of the king's path. A green dot marks the starting 1 square and a red dot the final square with number 855481. The red dot is surrounded by eight blue dots to show the occupied neighboring squares. A yellow dots marks the smallest unvisited square with number 2.

Cf. A333713 (choose lowest spiral number in case of tie), A335816, A316667, A330008, A329520, A326922, A328928, A328929, A033996.

A336038 Squares visited by a chess king on a square-spiral numbered board and stepping to the lowest unvisited adjacent square, where each step is not in the same direction as the previous step.

Original entry on oeis.org

1, 2, 3, 4, 6, 5, 15, 14, 12, 11, 9, 8, 22, 7, 19, 18, 16, 17, 35, 34, 60, 32, 13, 29, 28, 10, 25, 24, 46, 23, 45, 21, 20, 40, 39, 67, 37, 36, 38, 66, 64, 63, 97, 61, 62, 96, 95, 59, 33

Offset: 1

Scott R. Shannon, Jul 12 2020

This sequences gives the numbers of the squares visited by a chess king moving on a square-spiral numbered board where the king starts on the 1 numbered square and at each step, which is not in the same direction as its previous step, moves to an adjacent unvisited square, out of the eight adjacent neighboring squares, which contains the lowest spiral number.
The sequence is finite. After 48 steps the square with spiral number 33 is reached after which all eight adjacent squares have been visited.
If the king simply moved to the lowest numbered unvisited adjacent square the walk would be infinite as the king would just follow the path of the square spiral. By not allowing consecutive moves in the same direction forces the king off this minimal numbered path. The first time this happens is a(5) = 6 as from a(4) = 4 the lowest numbered adjacent square is 5 but that would require a step directly to the left, the same as the previous step from a(3) = 3 to a(4).

			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 of the king.
a(2) = 2. The eight adjacent unvisited squares around a(1) are numbered 2,3,4,5,6,7,8,9. Of these 2 is the lowest.
a(5) = 6. The five adjacent unvisited squares around a(4) = 4 are numbered 5,6,14,15,16. Of these 5 is the lowest but that would require a step directly left from 4, which is the same step as a(3) = 3 to a(4) = 4, so is not allowed. The next lowest available square is 6.

Scott R. Shannon, Image showing the 48 steps of the king's path. The green dot is the first square with number 1 and the red dot the last square with number 33. The red dot is surrounded by blue dots to show the eight occupied squares. The yellow dots marks the smallest unvisited square with number 26.
N. J. A. Sloane and Brady Haran, The Trapped Knight, Numberphile video (2019).

Cf. A336208, A335816, A333713, A333714, A316667.

A335856 Squares visited by a chess king on a spirally numbered infinite board where the king moves to the adjacent unvisited square containing the lowest prime number. If no such square is available it chooses the lowest-numbered adjacent unvisited square.

Original entry on oeis.org

1, 2, 3, 11, 29, 13, 31, 59, 32, 14, 4, 5, 17, 37, 67, 103, 149, 104, 66, 38, 18, 19, 7, 23, 47, 79, 48, 24, 8, 6, 20, 41, 71, 43, 73, 109, 72, 42, 21, 22, 44, 45, 46, 76, 75, 113, 74, 112, 110, 111, 157, 211, 271, 209, 269, 337, 267, 205, 151, 107, 69, 39, 40, 68, 105, 106, 70, 108

Offset: 1

Scott R. Shannon, Jun 27 2020

This sequences gives the numbers of the squares visited by a chess king moving on a square-spiral numbered board where the king starts on the 1 numbered square and at each step moves to the adjacent unvisited square containing the lowest prime number. If no adjacent unvisited square contains a prime number then the square with the lowest spiral number is chosen. Note that if the king simply moves to the lowest unvisited number the sequence will be infinite as the king will just follow the square spiral path.
The sequence is finite. After 719 steps the square with number 437 is visited, after which all adjacent neighboring squares have been visited.
Of the 719 visited squares 165 contain prime numbers while 554 contain composites. As the odd numbers are diagonally adjacent in the square spiral the king's path will contain many diagonal steps, often taking numerous diagonal steps is succession - see the attached link image.
The largest visited square is a(709) = 1367. The lowest unvisited square is 33.
The 719 steps until self-trapping occurs are significantly larger than the expected average of 210 moves to self-trapping for a random walk of the king on an infinite chessboard. See the link to the probability density graphs in A323562. - Hugo Pfoertner, Jul 19 2020
When the grid points are labeled starting with 0 at the origin, the king gets trapped after 171 moves at (3,0), after going as far as (10,-11) to the south-east and (-8,7) and (-5,8) to the north-east, see A383183. - M. F. Hasler, May 13 2025

			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 king.
a(2) = 2. The four unvisited squares around a(1) the king can move which contain prime numbers are 2,3,5,7. Of those 2 is the lowest.
a(4) = 11. The two unvisited squares around a(3) = 3 the king can move to which contain prime numbers are 11 and 13. Of those 11 is the lowest.
a(9) = 32. There are no unvisited squares around a(8) = 59 which contain prime numbers. The seven other unvisited squares are numbered 32,33,58,60,93,94,95. Of those 32 is the lowest.

M. F. Hasler, Table of n, a(n) for n = 1..720 (complete sequence), May 13 2025
Scott R. Shannon, Image showing the 719 steps of the path. A green dot marks the starting square and a red dot the final square with number 437. The red dot is surrounded by eight blue dots to show the occupied neighboring squares. A yellow dot marks the smallest unvisited square with number 33.

Cf. A333714, A333713, A335816, A336038, A336092, A316667.
Cf. A174344, A274923, A272763, A323561, A323562.
Cf. A000040 (the primes), A010051 (characteristic function of the primes).

from sympy import isprime # or use A010051
def square_number(z): return int(4*y**2-y-x if (y := z.imag) >= abs(x := z.real)
    else 4*x**2-x-y if -x>=abs(y) else (4*y-3)*y+x if -y>=abs(x) else (4*x-3)*x+y)
def A335856(n, moves=(1, 1+1j, 1j, 1j-1, -1, -1-1j, -1j, 1-1j)):
    if not hasattr(A:=A335856, 'terms'): A.terms=[1]; A.pos=0
    while len(A.terms) < n:
        try: move = min((1-isprime(s), s, z) for d in moves if
                        (s := square_number(z := A.pos+d)+1)not in A.terms)
        except ValueError:
            raise IndexError(f"Sequence has only {len(A.terms)} terms")
        A.terms.append(move[1]); A.pos = move[2]
    return A.terms[n-1]
A335856(999) # gives IndexError: Sequence has only 720 terms
A335856.terms # shows all 720 terms; append [:N] to see only N terms
# M. F. Hasler, May 13 2025

Name edited by Peter Munn, May 11 2025

More terms (complete sequence) from M. F. Hasler, May 13 2025

A336092 Squares visited by a chess king moving on a square-spiral numbered board where the king moves to the adjacent unvisited square containing the spiral number with the fewest divisors. In case of a tie it chooses the square with the largest spiral number.

Original entry on oeis.org

1, 7, 23, 47, 79, 49, 25, 9, 11, 29, 53, 87, 127, 177, 233, 299, 373, 454, 543, 641, 746, 859, 979, 1109, 1247, 1393, 1249, 1111, 983, 863, 751, 647, 753, 866, 865, 985, 1115, 1253, 1399, 1553, 1714, 1883, 2059, 2243, 2437, 2638, 2846, 3063, 3287, 3061, 2843, 2633, 2841, 3057, 3281, 3513, 3755

Offset: 1

Scott R. Shannon, Jul 08 2020

This sequences gives the numbers of the squares visited by a chess king moving on a square-spiral numbered board where the king starts on the 1 numbered square and at each step moves to an adjacent unvisited square, out of the eight adjacent neighboring squares, which contains the number with the fewest divisors. If two or more adjacent squares exist with the same fewest number of divisors then the square with the largest spiral number is chosen. Note that if the king simply moves to the largest available number the sequence will be infinite as the king will step along the south-east diagonal from square 1 forever.
The sequence is finite. After 21276 steps the square with spiral number 281747427 is visited, after which all adjacent neighboring squares have been visited. The end square is extremely far from the starting square, approximately 8860 units away, as the king is drawn generally outward due to its preference for the largest numbered square when the divisor counts are tied - see the link image. This end square spiral number is currently the largest for any square spiral single-visit trapped knight or trapped king path in the OEIS.
Due to the king's preference for squares with the fewest divisors it will move to a prime numbered square when possible, and the lowest prime if two or more unvisited primes are in adjacent squares. Of the 21276 visited squares 4363 contain prime numbers while 16913 contain composites. The largest visited square is a(21208) = 282486458.

			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 king.
a(2) = 7. The eight unvisited squares around a(1) the king can move to are numbered 2,3,4,5,6,7,8,9. Of these 2,3,5,7 have the minimum two divisors, and of those 7 is the largest.
a(3) = 23. The seven unvisited squares around a(2) the king can move to are numbered 6,8,19,20,21,22,23. Of these 19 and 23 have the minimum two divisors, and of those 23 is the largest.

Scott R. Shannon, Image showing the 21276 steps of the king's path. A green dot, far lowest left, marks the starting 1 square and a red dot, far upper right, the final square with number 281747427. The red dot is surrounded by eight blue dots to show the occupied neighboring squares. A yellow dots marks the smallest unvisited square with number 2. This is a high resolution image and may need to be downloaded to be viewed correctly.

Cf. A335816 (choose lowest number in case of tie), A333713, A333714, A316667, A330008, A329520, A326922.

A335900 Squares visited by a fairy chess wazir moving on a square-spiral numbered board where the wazir moves to the unvisited square containing the spiral number with the fewest divisors. In case of a tie it chooses the square with the lowest spiral number.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 23, 22, 21, 20, 19, 18, 17, 38, 37, 64, 65, 66, 67, 68, 39, 40, 41, 42, 43, 74, 73, 110, 109, 154, 155, 208, 269, 268, 337, 338, 339, 340, 271, 272, 211, 274, 275, 346, 347, 426, 427, 514, 515, 428, 349, 278, 277, 214, 159, 158, 157, 212, 213, 276

Offset: 1

Scott R. Shannon, Jun 29 2020

A fairy chess wazir can move one step in each of the four orthogonal grid directions, i.e., the same directions as a chess rook but only one square. In this sequence the wazir moves to the closest unvisited neighboring square which contains the number with the fewest divisors, and in case of a tie the square with the lowest spiral number. Note that if the wazir simply moves to the lowest available number the sequence will be infinite as the wazir will just follow the square spiral path.
The sequence is finite. After 61 steps the square with number 276 is visited, after which all four neighboring squares have been visited.
Due to the wazir's preference for squares with the fewest divisors it will move to a prime numbered square when possible, and the lowest prime if two or more unvisited primes are in neighboring squares. Of the 61 visited squares, 21 contain prime numbers, while 40 contain composites. The largest visited square is a(51) = 515.

			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 wazir.
a(2) = 2. The four unvisited squares around a(1) to which the wazir can move are numbered 2,4,6,8. Of these, 2 has only two divisors, so it is the square chosen.
a(9) = 23. The two unvisited squares around a(8) = 8 to which the wazir can move are numbered 9 and 23. Of these, 23 has only two divisors, so it is the square chosen.

Scott R. Shannon, Image showing the 61 steps of the wazir's path. A green dot marks the starting 1 square and a red dot the final square with number 276. The red dot is surrounded by four blue dots to show the unavailable neighboring squares. A yellow dot marks the smallest unvisited square with number 9.

Cf. A335816, A333713, A333714, A336092, A336038, A316667.

cp's OEIS Frontend

A333714 Squares visited by a chess king moving on a square-spiral numbered board where the king moves to the adjacent unvisited square containing the spiral number with the most divisors. In case of a tie it chooses the square with the highest spiral number.

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A336038 Squares visited by a chess king on a square-spiral numbered board and stepping to the lowest unvisited adjacent square, where each step is not in the same direction as the previous step.

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A335856 Squares visited by a chess king on a spirally numbered infinite board where the king moves to the adjacent unvisited square containing the lowest prime number. If no such square is available it chooses the lowest-numbered adjacent unvisited square.

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A336092 Squares visited by a chess king moving on a square-spiral numbered board where the king moves to the adjacent unvisited square containing the spiral number with the fewest divisors. In case of a tie it chooses the square with the largest spiral number.

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A335900 Squares visited by a fairy chess wazir moving on a square-spiral numbered board where the wazir moves to the unvisited square containing the spiral number with the fewest divisors. In case of a tie it chooses the square with the lowest spiral number.

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