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A327893 Minesweeper sequence of positive integers arranged in a hexagonal spiral. 1
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

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

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).

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.

EXAMPLE

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

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}]]

CROSSREFS

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

Sequence in context: A265273 A293770 A111311 * A326410 A255235 A293882

Adjacent sequences:  A327890 A327891 A327892 * A327894 A327895 A327896

KEYWORD

easy,sign

AUTHOR

Michael De Vlieger, Oct 09 2019

STATUS

approved

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Last modified February 19 13:03 EST 2020. Contains 332044 sequences. (Running on oeis4.)