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

1,1

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).

Links

Table of n, a(n) for n=1..80.

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.

Example

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.

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}]]

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).

Sequence in context: A309041 A249143 A121560 * A326408 A214717 A293312

Adjacent sequences: A326406 A326407 A326408 * A326410 A326411 A326412

Keyword

sign,tabl

Author

Witold Tatkiewicz, Oct 07 2019

Status

approved