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

%I A326410 #20 Mar 20 2020 23:36:57
%S A326410 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 A326410 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 A326410 -1,2,2,1,3,3,-1,1,2,3,-1,4,-1,3,2,0,1,2,-1,1,1
%N A326410 Minesweeper sequence of positive integers arranged on a square spiral on a 2D grid.
%C A326410 Place positive integers on a 2D grid starting with 1 in the center and continue along a spiral.
%C A326410 Replace each prime with -1 and each nonprime with the number of primes in adjacent grid cells around it.
%C A326410 n is replaced by a(n).
%C A326410 This sequence treats prime numbers as "mines" and fills gaps according to the rules of the classic Minesweeper game.
%C A326410 a(n) = 5 for n = 12.
%C A326410 Set of n such that a(n) = 4 is unbounded (conjecture).
%H A326410 Michael De Vlieger, <a href="/A326410/b326410.txt">Table of n, a(n) for n = 1..10201</a> (51 spiral iterations).
%H A326410 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 A326410 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 A326410 Wikipedia, <a href="https://en.wikipedia.org/wiki/Minesweeper_(video_game)">Minesweeper game</a>
%e A326410 Consider positive integers distributed onto the plane along the square spiral:
%e A326410 .
%e A326410   37--36--35--34--33--32--31
%e A326410    |                       |
%e A326410   38  17--16--15--14--13  30
%e A326410    |   |               |   |
%e A326410   39  18   5---4---3  12  29
%e A326410    |   |   |       |   |   |
%e A326410   40  19   6   1---2  11  28
%e A326410    |   |   |           |   |
%e A326410   41  20   7---8---9--10  27
%e A326410    |   |                   |
%e A326410   42  21--22--23--24--25--26
%e A326410    |
%e A326410   43--44--45--46--47--48--49--...
%e A326410 .
%e A326410 1 is not prime and in adjacent grid cells there are 4 primes: 2, 3, 5 and 7. Therefore a(1) = 4.
%e A326410 2 is prime, therefore a(2) = -1.
%e A326410 8 is not prime and in adjacent grid cells there are 4 primes: 2, 7 and 23. Therefore a(8) = 3.
%e A326410 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 A326410   *---2---2---1---3---3---*
%e A326410   |                       |
%e A326410   3   *---2---2---2---*   3
%e A326410   |   |               |   |
%e A326410   3   3   *---3---*   5   *
%e A326410   |   |   |       |   |   |
%e A326410   2   *   3   4---*   *   3
%e A326410   |   |   |           |   |
%e A326410   *   3   *---3---3---2   2
%e A326410   |   |                   |
%e A326410   3   3---2---*---2---1---.
%e A326410   |
%e A326410   *---1---1---2---*---2---1---...
%e A326410 In order to produce the sequence, the graph is read along the square spiral.
%Y A326410 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 A326410 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 A326410 Different arrangements of integers:
%Y A326410 Cf. A326405 (antidiagonals), A326406 (triangle maze), A326407 (square mapping), A326408 (square maze), A326409 (Hamiltonian path).
%K A326410 sign,tabl
%O A326410 1,1
%A A326410 _Witold Tatkiewicz_, Oct 07 2019