A383184 Diamond spiral numbers of the grid points visited by a king always moving to the unvisited point labeled with the smallest possible prime or else composite number.

0, 2, 3, 11, 23, 4, 5, 13, 12, 24, 41, 61, 40, 59, 83, 60, 84, 113, 85, 86, 62, 25, 26, 43, 14, 1, 7, 17, 31, 8, 19, 9, 10, 37, 21, 20, 53, 34, 33, 18, 32, 71, 97, 127, 72, 73, 50, 49, 48, 47, 29, 6, 15, 16, 30, 69, 68, 67, 28, 27, 44, 89, 64, 63, 42, 87, 88, 149, 116, 115, 114, 146, 223, 182, 181, 144, 179, 112, 111, 110, 109, 58, 38, 22, 57, 56, 79, 107, 139, 80, 81, 82, 39

Offset: 0

Author

M. F. Hasler, May 13 2025

Keywords

• nonn
• walk
• fini
• full

Comments

The infinite 2D grid is labeled along a diamond spiral as shown in A305258, starting with 0 at the origin (0,0), where each "shell" contains the points with given taxicab or L1-norm, as follows:

. (y)

2 | 8 17

| / \ \

1 | 9 2 7 16

| / / \ \ \

0 | 10 3 0--1 6 15

| \ \ / /

-1 | 11 4--5 14

| \ /

-2 | 12--13

|___________________

x: -2 -1 0 1 2 3

.

(This numbering, where the n-th "shell" has only 4n numbers, is "finer" than the square spiral numbering where the n-th shell has 8n numbers.)

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 92th move, the cursor is trapped in the point (-1,-3) labeled a(92) = 39. All eight neighbors were then already visited earlier, so the king has no more any possible move: see the "path plot" given in the links section.

Examples

			From the starting point (0,0) labeled a(0) = 0, the king can reach the point (0,1) 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,-1) labeled 11 = a(3), and so on.
The king reaches (1,7) and (1,-9) before getting trapped at (-1,-3) from where there is no more any unvisited point among the 8 neighbors.
		

Links

• M. F. Hasler, Path plot of A383184(0..92). (Dark blue points represent the 8 neighbors already visited around the red end point (-1,-3).)

Crossrefs

Cf. A383183 (same with square spiral numbering).

Cf. A305258 (more details about the diamond spiral).

Programs

Python

from sympy import isprime
def diamond_number(z):
    x, y = int(z.real), int(z.imag); d = abs(x)+abs(y)
    return 2*d*(d-1)+((x if y<0 else d+y)if x>0 else 2*d-x if y>0 else 3*d-y)
def A383184(n, moves=(1, 1+1j, 1j, 1j-1, -1, -1-1j, -1j, 1-1j)):
    if not hasattr(A:=A383184, '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 := diamond_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]
A383184(999) # gives IndexError: Sequence has only 93 terms
A383184.terms # shows the full sequence
import matplotlib.pyplot as plt # this and following to plot the path:
plt.plot([z.real for z in A383184.path], [z.imag for z in A383184.path])
plt.show()