A326405 Minesweeper sequence of positive integers arranged on a 2D grid along ascending antidiagonals.

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

Offset: 1

Author

Witold Tatkiewicz, Sep 26 2019

Keywords

• sign
• tabl

Comments

Map the positive integers on a 2D grid starting with 1 in top left corner and continue along increasing antidiagonals.

Replace each prime with -1 and each nonprime with the number of primes in adjacent grid cells around it.

If n is the original number, a(n) is the number that replaces it.

This sequence treats prime numbers as "mines" and fills gaps according to the rules of the classic Minesweeper game.

a(n) < 5 (conjectured).

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

Examples

			Consider positive integers distributed on the plane along antidiagonals:
   1  2  4  7 11 16 ...
   3  5  8 12 17 ...
   6  9 13 18 ...
  10 14 19 ...
  15 20 ...
  21 ...
  ...
1 is not prime and in its adjacent grid cells there are 3 primes: 2, 3 and 5. Therefore a(1) = 3.
2 is prime, therefore a(2) = -1.
8 is not prime and in adjacent grid cells there are 4 primes: 2, 5, 7 and 13. Therefore a(8) = 4.
From _Michael De Vlieger_, Oct 01 2019: (Start)
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:
  3  *  3  *  *  3  2  *  *  2  1  * ...
  *  *  4  4  *  *  3  3  *  2  1  2
  2  4  *  3  3  *  3  2  2  1  1  1
  .  2  *  3  2  2  3  *  3  1  1  *
  .  1  1  2  *  2  3  *  *  2  1  1
  .  1  1  2  3  *  3  3  *  3  1  .
  .  2  *  2  2  *  3  2  3  *  2  1
  .  2  *  2  1  1  2  *  2  1  3  *
  .  1  1  2  1  1  1  2  3  2  3  *
  .  1  1  2  *  2  1  1  *  *  2  2
  .  2  *  3  2  *  1  1  2  2  1  1
  .  2  *  3  2  2  1  1  1  1  .  1
   ...  (End)
		

Links

• Michael De Vlieger, Table of n, a(n) for n = 1..11325 (150 antidiagonals).
• 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

Different arrangements of integers:

Cf. A326406 - triangle maze,

Cf. A326407 - square mapping,

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[1 + PolygonalNumber[#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, Sep 30 2019 *)