A326407 Minesweeper sequence of positive integers arranged on a 2D grid along a square array that grows by alternately adding a row at its bottom edge and a column at its right edge.

2, -1, -1, 2, -1, 5, -1, 2, 1, 3, -1, 4, -1, 3, 2, 0, -1, 3, -1, 3, 3, 2, -1, 1, 0, 2, 3, 2, -1, 3, -1, 1, 2, 2, 2, 0, -1, 1, 2, 3, -1, 3, -1, 3, 3, 2, -1, 1, 0, 1, 2, 2, -1, 2, 3, 2, 3, 2, -1, 2, -1, 3, 2, 0, 1, 2, -1, 2, 1, 1, -1, 3, -1, 1, 1, 1, 3, 3, -1, 1, 0

Offset: 1

Author

Witold Tatkiewicz, Oct 02 2019

Keywords

• sign
• tabl

Comments

Place positive integers on a 2D grid starting with 1 in the top left corner and continue along an increasing square array as in A060734.

Replace each prime with -1 and each nonprime with 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 the rules of the classic Minesweeper game.

a(n) < 6 (conjectured).

a(n) = 5 for n={6} (conjectured).

Set of n such that a(n) = 4 is unbounded (conjectured).

Examples

			Consider positive integers distributed onto the plane along an increasing square array:
   1  4  9 16 25 36
   2  3  8 15 24 35
   5  6  7 14 23 34
  10 11 12 13 22 33
  17 18 19 20 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 2 primes: 3, and 7. Therefore a(8) = 2.
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  1  .  .  .  .  .  .  .  .  . ...
  *  *  2  2  1  2  1  2  1  1  .  1
  *  5  *  3  *  2  *  3  *  2  1  1
  3  *  4  *  2  2  2  *  3  *  1  1
  *  3  *  3  3  1  3  2  3  1  2  1
  2  3  2  *  3  *  3  *  1  .  1  *
  *  1  2  3  *  3  *  2  1  .  2  3
  1  2  2  *  2  3  2  3  1  2  2  *
  1  2  *  2  1  1  *  3  *  2  *  2
  2  *  3  2  .  2  3  *  3  3  1  1
  *  3  *  1  1  2  *  3  *  2  1  .
  1  2  1  2  2  *  3  3  2  *  1  1
...
In order to produce the sequence, the graph is read along the 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. A060734 - plane mapping

Different arrangements of integers:

Cf. A326405 - antidiagonals,

Cf. A326406 - triangle maze,

Cf. A326408 - square maze,

Cf. A326409 - Hamiltonian path,

Cf. A326410 - Ulam's spiral.

Programs

Java

See Links section.

Mathematica

Block[{n = 12, s}, s = ArrayPad[Array[If[#1 < 2 #2 - 1, #2^2 + #2 - #1, (#1 - #2)^2 + #2] & @@ {#1 + #2 - 1, #2} &, {# + 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, PolygonalNumber@ n}]] (* _Michael De Vlieger, Oct 02 2019 *)