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A326410 Minesweeper sequence of positive integers arranged on square spiral on 2d grid. 7
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, 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, -1, 2, 2, 1, 3, 3, -1, 1, 2, 3, -1, 4, -1, 3, 2, 0, 1, 2, -1, 1, 1 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

Place positive integers on 2-d grid starting with 1 in the centre and continue along spiral.

Replace primes by -1 and nonprimes by the number of primes in adjacent grid cells around them.

n is replaced by a(n).

This sequence treats prime numbers as "mines" and fills gaps according to classical Minesweeper game.

a(n) = 5 for n = 12.

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

LINKS

Michael De Vlieger, Table of n, a(n) for n = 1..10201 (51 spiral iterations).

Michael De Vlieger, Minesweeper-style graph read along original mapping, replacing -1 with a "mine", and 0 with blank space.

Michael De Vlieger, Square plot of 10^3 spiral iterations read along original mapping, with black indicating a prime and levels of gray commensurate to a(n).

Wikipedia, Minesweeper game

EXAMPLE

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

.

  37--36--35--34--33--32--31

   |                       |

  38  17--16--15--14--13  30

   |   |               |   |

  39  18   5---4---3  12  29

   |   |   |       |   |   |

  40  19   6   1---2  11  28

   |   |   |           |   |

  41  20   7---8---9--10  27

   |   |                   |

  42  21--22--23--24--25--26

   |

  43--44--45--46--47--48--49--...

.

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

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

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

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:

  *---2---2---1---3---3---*

  |                       |

  3   *---2---2---2---*   3

  |   |               |   |

  3   3   *---3---*   5   *

  |   |   |       |   |   |

  2   *   3   4---*   *   3

  |   |   |           |   |

  *   3   *---3---3---2   2

  |   |                   |

  3   3---2---*---2---1---.

  |

  *---1---1---2---*---2---1---...

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

CROSSREFS

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

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

Different arrangements of integers:

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

Sequence in context: A293770 A111311 A327893 * A255235 A293882 A016524

Adjacent sequences:  A326407 A326408 A326409 * A326411 A326412 A326413

KEYWORD

sign,tabl

AUTHOR

Witold Tatkiewicz, Oct 07 2019

STATUS

approved

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Last modified January 20 19:53 EST 2020. Contains 331096 sequences. (Running on oeis4.)