cp's OEIS Frontend

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.

%I A383184 #8 May 24 2025 17:53:58
%S A383184 0,2,3,11,23,4,5,13,12,24,41,61,40,59,83,60,84,113,85,86,62,25,26,43,
%T A383184 14,1,7,17,31,8,19,9,10,37,21,20,53,34,33,18,32,71,97,127,72,73,50,49,
%U A383184 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
%N 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.
%C A383184 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:
%C A383184 . (y)
%C A383184   2 |        8  17
%C A383184     |      /  \   \
%C A383184   1 |     9  2  7  16
%C A383184     |   /  /   \  \  \
%C A383184   0 | 10  3  0--1  6  15
%C A383184     |   \  \      /  /
%C A383184  -1 |    11  4--5  14
%C A383184     |      \      /
%C A383184  -2 |       12--13
%C A383184     |_____________________
%C A383184  x:   -2 -1  0  1  2  3
%C A383184 .
%C A383184 (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.)
%C A383184 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.
%C A383184 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.
%H A383184 M. F. Hasler, <a href="/A383184/a383184.png">Path plot of A383184(0..92)</a>. (Dark blue points represent the 8 neighbors already visited around the red end point (-1,-3).)
%e A383184 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.
%e A383184 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.
%e A383184 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.
%o A383184 (Python)
%o A383184 from sympy import isprime
%o A383184 def diamond_number(z):
%o A383184     x, y = int(z.real), int(z.imag); d = abs(x)+abs(y)
%o A383184     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)
%o A383184 def A383184(n, moves=(1, 1+1j, 1j, 1j-1, -1, -1-1j, -1j, 1-1j)):
%o A383184     if not hasattr(A:=A383184, 'terms'): A.terms=[0]; A.pos=0; A.path=[0]
%o A383184     while len(A.terms) <= n:
%o A383184         try: _,s,z = min((1-isprime(s), s, z) for d in moves if
%o A383184                          (s := diamond_number(z := A.pos+d))not in A.terms)
%o A383184         except ValueError:
%o A383184             raise IndexError(f"Sequence has only {len(A.terms)} terms")
%o A383184         A.terms.append(s); A.pos = z; A.path.append(z)
%o A383184     return A.terms[n]
%o A383184 A383184(999) # gives IndexError: Sequence has only 93 terms
%o A383184 A383184.terms # shows the full sequence
%o A383184 import matplotlib.pyplot as plt # this and following to plot the path:
%o A383184 plt.plot([z.real for z in A383184.path], [z.imag for z in A383184.path])
%o A383184 plt.show()
%Y A383184 Cf. A383183 (same with square spiral numbering).
%Y A383184 Cf. A305258 (more details about the diamond spiral).
%K A383184 nonn,walk,fini,full
%O A383184 0,2
%A A383184 _M. F. Hasler_, May 13 2025