A326410 Minesweeper sequence of positive integers arranged on a square spiral on a 2D grid.

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

Offset: 1

Author

Witold Tatkiewicz, Oct 07 2019

Keywords

• sign
• tabl

Comments

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

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

n is replaced by a(n).

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

a(n) = 5 for n = 12.

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

Examples

			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, the graph is read along the square spiral.
		

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

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).