A327893 Minesweeper sequence of positive integers arranged in a hexagonal spiral.

4, -1, -1, 3, -1, 3, -1, 3, 2, 4, -1, 3, -1, 3, 2, 2, -1, 3, -1, 2, 0, 2, -1, 3, 1, 2, 2, 2, -1, 2, -1, 1, 1, 1, 2, 3, -1, 2, 0, 1, -1, 4, -1, 1, 1, 2, -1, 1, 0, 1, 2, 3, -1, 3, 1, 0, 0, 1, -1, 4, -1, 1, 0, 0, 1, 3, -1, 2, 2, 2, -1, 2, -1, 3, 1, 1, 0, 1, -1, 3

Offset: 1

Author

Michael De Vlieger, Oct 09 2019

Keywords

• easy
• sign

Comments

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

The largest term in the sequence is 4 since 1 is surrounded by 3 odd numbers {3, 5, 7} and the only even prime. Additionally, the pattern of odd and even numbers appears in alternating rows oriented in a triangular symmetry such that no other number has more than four odd numbers. (This courtesy of Witold Tatkiewicz.)

Examples

			Consider a spiral grid drawn counterclockwise with the largest number k = A003219(n) = 3*n*(n+1)+1 in "shell" n, and each shell has A008458(n) elements:
          28--27--26--25
          /             \
        29  13--12--11  24
        /   /         \   \
      30  14   4---3  10  23
      /   /   /     \   \   \
    31  15   5   1---2   9  22
      \   \   \         /   /
      32  16   6---7---8  21
        \   \             /
        33  17--18--19--20  ...
          \                /
          34--35--36--37--38
1 is not prime and in the 6 adjacent cells 2 through 7 inclusive, we have 4 primes, therefore a(1) = 4.
2 is prime therefore a(2) = -1.
4 is not prime and in the 6 adjacent cells {1, 3, 12, 13, 14, 5} there are 4 primes, therefore a(4) = 4, etc.
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---2---1
        /             \
       *   *---3---*   3
      /   /         \   \
     2   3   3---*   4   *
    /   /   /     \   \   \
   *   2   *   4---*   2   2
    \   \   \         /   /
     1   3   3---*---3   .
      \   \             /
       1   *---3---*---2  ...
        \                 /
         1---2---3---*---2
		

Links

• Michael De Vlieger, Table of n, a(n) for n = 1..10621 (60 concentric hexagons).
• Michael De Vlieger, Minesweeper style hexagonal plot of 1261 terms, replacing -1 with n in a black circle, and 0 represented by blank space.
• Michael De Vlieger, Hexagonal plot of 30301 terms, (100 concentric hexagons), color coded.
• Michael De Vlieger, Hexagonal plot of 120601 terms, (200 concentric hexagons), color coded.
• Michael De Vlieger, Plot of 469 terms, with 12 concentric hexagons smoothed into concentric rings, color coded.
• Michael De Vlieger, Plot of 120601 terms, with 200 concentric hexagons smoothed into concentric rings, color coded.

Crossrefs

Cf. A003215, A008458, A326405, A326406, A326407, A326408, A326409, A326410.

Programs

Mathematica

Block[{n = 6, m, s, t, u}, m = n + 1; s = Array[3 #1 (#1 - 1) + 1 + #2 #1 + #3 & @@ {#3, #4, Which[Mod[#4, 3] == 0, Abs[#1], Mod[#4, 3] == 1, Abs[#2], True, Abs[#2] - Abs[#1]]} & @@ {#1, #2, If[UnsameQ @@ Sign[{#1, #2}], Abs[#1] + Abs[#2], Max[Abs[{#1, #2}]]], Which[And[#1 > 0, #2 <= 0], 0, And[#1 >= #2, #1 + #2 > 0], 1, And[#2 > #1, #1 >= 0], 2, And[#1 < 0, #2 >= 0], 3, And[#1 <= #2, #1 + #2 < 0], 4, And[#1 > #2, #1 + #2 <= 0], 5, True, 0]} & @@ {#2 - m - 1, m - #1 + 1} &, {#, #}] &[2 m + 1]; t = s /. k_ /; k > 3 n (n + 1) + 1 :> -k; Table[If[PrimeQ@ m, -1, Count[#, _?PrimeQ] &@ Union@ Map[t[[#1, #2]] & @@ # &, Join @@ Array[FirstPosition[t, m] + {##} - 2 & @@ {#1, #2 + Boole[#1 == #2 == 2] + Boole[#1 == 1]} &, {3, 2}]]], {m, 3 n (n - 1) + 1}]]