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A327893
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Minesweeper sequence of positive integers arranged in a hexagonal spiral.
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1
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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
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OFFSET
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1,1
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COMMENTS
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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.)
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LINKS
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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.
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EXAMPLE
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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
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MATHEMATICA
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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}]]
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CROSSREFS
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KEYWORD
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easy,sign
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AUTHOR
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STATUS
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approved
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