A335856 -id:A335856 - cp's OEIS frontend

A383185 Number of the square visited by a king moving on a spirally numbered board always to the lowest available unvisited square, when a wall delimiting the spiral must be crossed on each move.

0, 3, 13, 2, 10, 1, 7, 21, 6, 18, 4, 14, 32, 12, 28, 11, 27, 9, 23, 8, 22, 44, 20, 40, 19, 5, 17, 37, 16, 34, 15, 33, 59, 31, 57, 30, 54, 29, 53, 85, 51, 25, 47, 24, 46, 76, 45, 75, 43, 73, 42, 70, 41, 69, 39, 67, 38, 66, 36, 62, 35, 61, 95, 60, 94, 58, 92, 56, 88, 55, 87, 127, 86, 52, 26

Offset: 0

M. F. Hasler, May 12 2025

The board is numbered following a square spiral starting with 0 at the origin (where the king is at n = 0) and delimited by a wall that must be crossed on each move:
.
  16  15  14  13  12 | .
     ,-----------.   | .
  17 | 4   3   2 |11 | .
     |   ,---    |   | .
  18 | 5 | 0   1 |10 | .
     |   '-------'   | .
  19 | 6   7   8   9 | .
     `---------------' .
  20  21  22  23  24  25
.
A line drawn from the center of the starting square to the center of the ending square must pass through a wall on each move. A move that would just touch a wall without passing through the wall (e.g., 0 to 2) is not permissible. Equivalently, the king can't move from a square labeled k to a square labeled k +- 1 or k +- 2, i.e., |a(n)-a(n+1)| > 2 for all n.
This sequence is a permutation of the nonnegative integers, see A383186 for the inverse permutation. The king's walk indeed fills the 2D grid with an initial segment S0 of 24 moves, followed by rings R(r), r >= 1, which consist of three shells S1(r), S2(r) and S3(r), each of which corresponds to a tour around the center. Each ring R(r) starts with the move number n = 48 r^2 - 16 r + 1 = (25, 147, 365, ...) to the square at position P(r) = (2-3r, 3r-3) = ((-1,0), (-4,3), (-7,6), ...), and contains a perfectly well defined sequence of 96 r + 26 grid points following a precise sequence of pattern given in full detail on the wiki page provided in the links section.

			For n = 1, a(1) = 3 because moving from 0 to 1 or 2 does not pass through a wall.

M. F. Hasler, Table of n, a(n) for n = 0..1000
M. F. Hasler, A383185: Details about the grid filling king's walk, OEIS wiki, May 15, 2025.
M. F. Hasler, Path plot with annotations, May 15, 2025.
OEIS index to sequences related to permutations of the integers.

Cf. A375925 (the same with indices and numbers of squares starting at 1).
Cf. A383186 (inverse permutation).
Cf. A316328 (knight's path), A033638, A316667 (trapped knight), A336038 (trapped king), A335856 (trapped king preferably moving to prime numbers).

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 A383185(n):
    if not hasattr(A:=A383185, 'terms'): A.terms=[0]; A.pos=0; A.path=[0]
    while len(A.terms) <= n:
        s,d = min((s,d) for d in (1, 1+1j, 1j, 1j-1, -1, -1-1j, -1j, 1-1j) if
            abs((s:=square_number(A.pos+d))-A.terms[-1]) > 2 and s not in A.terms)
        A.terms.append(s); A.pos += d; A.path.append(A.pos)
    return A.terms[n]
import matplotlib.pyplot as plt # this and below to plot the trajectory
plt.plot([z.real for z in A383185.path], [z.imag for z in A383185.path])
plt.show()

A336494 The number of steps for a walk on a square spiral numbered board when starting on square 1 and stepping to an unvisited square containing the lowest prime number, where the square is within a block of size (2n+1) X (2n+1) centered on the current square. If no unvisited prime numbered squares exist within the block the walk ends.

Original entry on oeis.org

7, 37, 65, 308, 654, 7214, 21992, 49850, 222791, 1146922, 1912101, 6372680, 23077800

Offset: 1

Scott R. Shannon, Jul 23 2020

For n = 1 this sequence is similar to A335856 except that only prime numbers can be stepped to; if no adjacent prime number exists then the walk ends. In general for a(n) the walk can step to any unvisited square containing the lowest prime number within a block of size (2n+1) X (2n+1) centered on the current square.

See A336576 for the final square number of the walks.

			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) = 7. Starting from the square 1 the sequence of adjacent unvisited lowest primes the walk can step to are 2,3,11,29,13,31,59. Once the square 59 is visited there are no other unvisited adjacent squares containing primes, so the walk terminates after 7 steps. See the first linked image.
a(2) = 38. This walk also starts by stepping to 2 and then 3. But the next lowest prime 5 is now two units away so is reachable and is thus the next stepped to square. Further steps are 7,19,17,37...,827,829,719,947. Once the square 947 is visited there are no other unvisited squares containing primes within the surrounding 5x5 block of squares, so the walk terminates after 38 steps. See the second linked image.
Also see the linked images for n=3,4,5,6.

Scott R. Shannon, Image showing the 7 steps of the walk for n = 1. For this and other images a green square shows the starting 1 square, a red square shows the final square, 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. Blue squares mark out the (2n+1)x(2n+1) block of squares around the final square which contains no further unvisited primes. The square spiral numbering of the board is shown as a thin white line.
Scott R. Shannon, Image showing the 37 steps of the walk for n = 2. Click on this and other images to zoom in.
Scott R. Shannon, Image showing the 65 steps of the walk for n = 3.
Scott R. Shannon, Image showing the 308 steps of the walk for n = 4.
Scott R. Shannon, Image showing the 654 steps of the walk for n = 5.
Scott R. Shannon, Image showing the 7214 steps of the walk for n = 6.
Wikipedia, Ulam Spiral.

Cf. A336576 (final square number), A335856, A000040, A136626, A336092, A330979, A332767, A335661, A335364.

A383183 Square spiral numbers of the n-th grid point visited by a king always moving to the unvisited point labeled with the smallest possible prime or else composite number.

Original entry on oeis.org

0, 2, 3, 5, 7, 23, 47, 79, 48, 24, 8, 1, 11, 13, 31, 29, 53, 27, 9, 10, 26, 25, 49, 83, 50, 51, 52, 28, 12, 30, 54, 55, 89, 131, 179, 129, 87, 127, 85, 84, 124, 173, 229, 293, 227, 169, 223, 167, 119, 80, 81, 82, 122, 120, 121, 168, 170, 171, 123, 172, 228, 292, 226, 224, 225, 287, 359, 439

Offset: 0

M. F. Hasler, May 13 2025

The infinite 2D grid is labeled along a square spiral as shown in A316328, starting with 0 at the origin (0,0), where the n-th shell contains the 8n points with sup norm n, as follows:
.
  16--15--14--13--12   :
   |               |   :
  17   4---3---2  11  28
   |   |       |   |   |
  18   5   0---1  10  27
   |   |           |   |
  19   6---7---8---9  26
   |                   |
  20--21--22--23--24--25
.
The cursor is moving like a chess king to the von Neumann neighbor not visited earlier and labeled with the smallest prime number if possible, otherwise with the smallest possible composite number.
After the 171th move, the cursor is trapped in the point (3,0) labeled a(171) = 33. All eight neighbors were then already visited earlier, so the king has no more any possible move.

			From the starting point (0,0) labeled a(0) = 0, the king can reach the point (1,1) labeled 2, which is the smallest possible prime number, so a(1) = 2.
Then the king can reach (1,0) labeled 3 which is the next smaller prime number, so a(2) = 3. From there it can go to (-1,0) labeled 5 = a(3), and then to (0,-1) labeled a(4) 7 = a(4). From there, the only available prime number is 23 = a(5) at (1,-2).
The king continues in that south-east direction, before walking in north-east direction and then back and further south-east up to and beyond the point (10,-10). Then it goes back in the opposite north-west direction up to (-8,7) and (-5, 8), before heading to the point (3,0) where it gets stuck.

M. F. Hasler, Table of n, a(n) for n = 0..171 (full sequence).
M. F. Hasler, Path plot of A383183(0..171). (The 8 dark blue points around the red ending point represent the 8 already visited neighbors.) May 13, 2025.

Cf. A383184 (the same with "diamond spiral" numbering).
Cf. A383185 (similar with |a(n)-a(n+1)| > 2 instead of the prime number condition).
Cf. A335856 (same with the square spiral and indices starting at 1).
Cf. A316328 (knight path on square spiral numbered board).

from sympy import isprime # = 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 A383183(n, moves=(1, 1+1j, 1j, 1j-1, -1, -1-1j, -1j, 1-1j)):
    if not hasattr(A:=A383183, 'terms'): A.terms=[0]; A.pos=0; A.path=[0]
    while len(A.terms) <= n:
        try: _,s,z = min((1-isprime(s), s, z) for d in moves if
                     (s := square_number(z := A.pos+d)) not in A.terms)
        except ValueError:
            raise IndexError(f"Sequence has only {len(A.terms)} terms")
        A.terms.append(s); A.pos = z; A.path.append(z)
    return A.terms[n]
A383183(999) # gives IndexError: Sequence has only 172 terms
A383183.terms # shows the full sequence; append [:N] to show only N terms
import matplotlib.pyplot as plt # this and following to plot the path:
plt.plot([z.real for z in A383183.path], [z.imag for z in A383183.path])
plt.show()

A336576 The final square number for a walk on a square spiral numbered board when starting on square 1 and stepping to an unvisited square containing the lowest prime number, where the square is within a block of size (2n+1) x (2n+1) centered on the current square. If no unvisited prime numbered squares exist within the block the walk ends.

Original entry on oeis.org

59, 947, 313, 3331, 5659, 67547, 253801, 676259, 3162413, 16604417, 29135971, 108235159, 437456497

Offset: 1

Scott R. Shannon, Jul 26 2020

See A336494 for an explanation of the sequence and images of the walks.

			a(1) = 59. Starting from the square 1 the sequence of adjacent unvisited lowest primes the walk can step to are 2,3,11,29,13,31,59. Once the square 59 is visited there are no other unvisited adjacent squares containing primes, so the walk terminates.

Wikipedia, Ulam Spiral.

Cf. A336494 (total number of steps), A335856, A000040, A136626, A336092, A330979, A332767, A335661, A335364.

cp's OEIS Frontend

A383185 Number of the square visited by a king moving on a spirally numbered board always to the lowest available unvisited square, when a wall delimiting the spiral must be crossed on each move.

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A383183 Square spiral numbers of the n-th grid point visited by a king always moving to the unvisited point labeled with the smallest possible prime or else composite number.

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