A326409 Minesweeper sequence of positive integers arranged on a 2D grid along Hamiltonian path.

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

Offset: 1

Author

Witold Tatkiewicz, Oct 07 2019

Keywords

• sign
• tabl

Comments

Place positive integers on a 2D grid starting with 1 in the top left corner and continue along Hamiltonian path A163361 or A163363.

Replace each prime with -1 and each nonprime by 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 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 an increasing Hamiltonian path (in this case it starts downwards):
.
   1   4---5---6  59--60--61  64--...
   |   |       |   |       |   |
   2---3   8---7  58--57  62--63
           |           |
  15--14   9--10  55--56  51--50
   |   |       |   |       |   |
  16  13--12--11  54--53--52  49
   |                           |
  17--18  31--32--33--34  47--48
       |   |           |   |
  20--19  30--29  36--35  46--45
   |           |   |           |
  21  24--25  28  37  40--41  44
   |   |   |   |   |   |   |   |
  22--23  26--27  38--39  42--43
.
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 3 primes: 5, 3 and 7. Therefore a(8) = 3.
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---*---3   *---2---*   1 ...
  |   |       |   |       |   |
  *---*   3---*   2---2   1---1
          |           |
  3---3   4---2   3---1   1---.
  |   |       |   |       |   |
  2   *---3---*   2---*---2   1
  |                           |
  *---4   *---3---3---2   *---1
      |   |           |   |
  2---*   3---*   2---3   2---2
  |           |   |           |
  2   2---3   2   *   2---*   2
  |   |   |   |   |   |   |   |
  1---*   1---1   1---2   2---*
In order to produce the sequence, the graph is read along its original mapping.
		

Links

• Alexander Bogomolny, Plane Filling Curves: Hilbert's & Moore's, Cut the Knot.org, retrieved October 2019.
• Eric Weisstein's World of Mathematics, Hilbert Curve.
• Wikipedia, Hilbert Curve.
• Wikipedia, Minesweeper game.

Crossrefs

Cf. A163361 (plane mapping), A163363 (alternative plane mapping).

Different arrangements of integers: A326405 (antidiagonals), A326406 (triangle maze), A326407 (square mapping), A326408 (square maze), A326410 (Ulam's spiral).

Programs

Mathematica

Block[{nn = 4, s, t, u}, s = ConstantArray[0, {2^#, 2^#}] &[nn + 1]; t = First[HilbertCurve@ # /. Line -> List] &[nn + 1] &[nn + 1]; s = ArrayPad[ReplacePart[s, Array[{1, 1} + t[[#]] -> # &, 2^(2 (nn + 1))]], {{1, 0}, {1, 0}}]; u = Table[If[PrimeQ@ m, -1, Count[#, _?PrimeQ] &@ Union@ Map[s[[#1, #2]] & @@ # &, Join @@ Array[FirstPosition[s, m] + {##} - 2 &, {3, 3}]]], {m, (2^nn)^2}]]