A326408 Minesweeper sequence of positive integers arranged on a 2D grid along a square maze.

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

Offset: 1

Author

Witold Tatkiewicz, Oct 04 2019

Keywords

• sign
• tabl

Comments

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).

Examples

			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.
		

Links

• Michael De Vlieger, Table of n, a(n) for n = 1..10000
• 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).
• Witold Tatkiewicz, link for Java program
• Wikipedia, Minesweeper game

Crossrefs

Cf. A081344 - plane mapping

Different arrangements of integers:

Cf. A326405 - antidiagonals,

Cf. A326406 - triangle maze,

Cf. A326407 - square mapping,

Cf. A326409 - Hamiltonian path,

Cf. A326410 - Ulam's spiral.

Programs

Java

// See Links section.

Mathematica

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 *)