A326410 - OEIS

%I #20 Mar 20 2020 23:36:57

%S 4,-1,-1,3,-1,3,-1,3,3,2,-1,5,-1,2,2,2,-1,3,-1,3,3,2,-1,2,1,0,2,3,-1,

%T 3,-1,3,3,1,2,2,-1,3,3,2,-1,3,-1,1,1,2,-1,2,1,1,1,1,-1,2,3,2,2,2,-1,2,

%U -1,2,2,1,3,3,-1,1,2,3,-1,4,-1,3,2,0,1,2,-1,1,1

%N Minesweeper sequence of positive integers arranged on a square spiral on a 2D grid.

%C Place positive integers on a 2D grid starting with 1 in the center and continue along a spiral.

%C Replace each prime with -1 and each nonprime with the number of primes in adjacent grid cells around it.

%C n is replaced by a(n).

%C This sequence treats prime numbers as "mines" and fills gaps according to the rules of the classic Minesweeper game.

%C a(n) = 5 for n = 12.

%C Set of n such that a(n) = 4 is unbounded (conjecture).

%H Michael De Vlieger, <a href="/A326410/b326410.txt">Table of n, a(n) for n = 1..10201</a> (51 spiral iterations).

%H Michael De Vlieger, <a href="/A326410/a326410.png">Minesweeper-style graph</a> read along original mapping, replacing -1 with a "mine", and 0 with blank space.

%H Michael De Vlieger, <a href="/A326410/a326410_1.png">Square plot of 10^3 spiral iterations</a> read along original mapping, with black indicating a prime and levels of gray commensurate to a(n).

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Minesweeper_(video_game)">Minesweeper game</a>

%e Consider positive integers distributed onto the plane along the square spiral:

%e .

%e   37--36--35--34--33--32--31

%e    |                       |

%e   38  17--16--15--14--13  30

%e    |   |               |   |

%e   39  18   5---4---3  12  29

%e    |   |   |       |   |   |

%e   40  19   6   1---2  11  28

%e    |   |   |           |   |

%e   41  20   7---8---9--10  27

%e    |   |                   |

%e   42  21--22--23--24--25--26

%e    |

%e   43--44--45--46--47--48--49--...

%e .

%e 1 is not prime and in adjacent grid cells there are 4 primes: 2, 3, 5 and 7. Therefore a(1) = 4.

%e 2 is prime, therefore a(2) = -1.

%e 8 is not prime and in adjacent grid cells there are 4 primes: 2, 7 and 23. Therefore a(8) = 3.

%e Replacing n with a(n) in the plane described above, and using "." for a(n) = 0 and "*" for negative a(n), we produce a graph resembling Minesweeper, where the mines are situated at prime n:

%e   *---2---2---1---3---3---*

%e   |                       |

%e   3   *---2---2---2---*   3

%e   |   |               |   |

%e   3   3   *---3---*   5   *

%e   |   |   |       |   |   |

%e   2   *   3   4---*   *   3

%e   |   |   |           |   |

%e   *   3   *---3---3---2   2

%e   |   |                   |

%e   3   3---2---*---2---1---.

%e   |

%e   *---1---1---2---*---2---1---...

%e In order to produce the sequence, the graph is read along the square spiral.

%Y Cf. A136626 - similar sequence: For every number n in Ulam's spiral the sequence gives the number of primes around it (number n excluded).

%Y Cf. A136627 - similar sequence: For every number n in Ulam's spiral the sequence gives the number of primes around it (number n included).

%Y Different arrangements of integers:

%Y Cf. A326405 (antidiagonals), A326406 (triangle maze), A326407 (square mapping), A326408 (square maze), A326409 (Hamiltonian path).

%K sign,tabl

%O 1,1

%A _Witold Tatkiewicz_, Oct 07 2019