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A326408
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Minesweeper sequence of positive integers arranged on a 2D grid along a square maze.
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7
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2, -1, -1, 3, -1, 3, -1, 4, 3, 1, -1, 4, -1, 3, 4, 2, -1, 2, -1, 2, 3, 3, -1, 2, 1, 0, 2, 3, -1, 2, -1, 2, 2, 1, 3, 2, -1, 1, 1, 2, -1, 4, -1, 2, 3, 3, -1, 1, 0, 0, 2, 3, -1, 1, 1, 1, 3, 3, -1, 3, -1, 2, 2, 1, 0, 1, -1, 3, 3, 2, -1, 2, -1, 2, 1, 0, 1, 2, -1, 2, 1
(list;
table;
graph;
refs;
listen;
history;
text;
internal format)
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OFFSET
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1,1
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COMMENTS
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Place positive integers on 2D grid starting with 1 in the top left corner and continue along the square maze as in A081344.
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.
Set of n such that a(n) = 4 is unbounded (conjectured).
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LINKS
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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 a million terms read along original mapping, with black indicating a prime and levels of gray commensurate to a(n).
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EXAMPLE
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Consider positive integers distributed onto the plane along increasing square array:
1 4 5 16 17 36 ...
2 3 6 15 18 35
9 8 7 14 19 34
10 11 12 13 20 33
25 24 23 22 21 32
26 27 28 29 30 31
...
1 is not prime and in adjacent grid cells there are 2 primes: 2 and 3. Therefore a(1) = 2.
2 is prime, therefore a(2) = -1.
8 is not prime and in adjacent grid cells there are 4 primes: 2, 3, 7 and 11. Therefore a(8) = 4.
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 3 * 2 * 2 * 1 . 1 * 1 ...
* * 3 4 2 3 1 2 1 3 2 2
3 4 * 3 * 1 1 2 * 3 * 1
1 * 4 * 2 2 2 * 3 * 2 2
1 2 * 3 3 2 * 3 3 1 2 2
. 2 3 * 2 * 4 * 2 2 2 *
. 1 * 3 3 2 * 3 * 2 * 4
. 2 3 * 1 1 1 3 2 4 3 *
1 2 * 2 1 . 1 2 * 2 * 2
1 * 2 1 . . 1 * 3 3 1 1
1 1 1 . 1 1 2 2 * 2 1 .
. 1 1 1 1 * 2 2 2 * 1 1
...
In order to produce the sequence, the graph is read along its original mapping.
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MATHEMATICA
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Block[{n = 9, s}, s = ArrayPad[Array[If[#1 < 2 #2 - 1, #2^2 + #2 - #1, (#1 - #2)^2 + #2] & @@ {#1 + #2 - 1, #2} & @@ If[Or[And[#2 < #1, EvenQ@ #1], And[#1 < #2, EvenQ@ #2]], {#1, #2}, {#2, #1}] &, {# + 1, # + 1}], 1] &@ n; Table[If[PrimeQ@ m, -1, Count[#, _?PrimeQ] &@ Union@ Map[s[[#1, #2]] & @@ # &, Join @@ Array[FirstPosition[s, m] + {##} - 2 &, {3, 3}]]], {m, n^2}] (* Michael De Vlieger, Oct 04 2019 *)
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PROG
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(Java) See Links section.
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CROSSREFS
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Different arrangements of integers:
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KEYWORD
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AUTHOR
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STATUS
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approved
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