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A326406 Minesweeper sequence of positive integers arranged on a 2D grid along a triangular maze. 7
3, -1, -1, 2, -1, 3, -1, 4, 4, 1, -1, 3, -1, 3, 2, 1, -1, 3, -1, 3, 2, 1, -1, 2, 3, 2, 3, 1, -1, 3, -1, 2, 2, 1, 2, 1, -1, 2, 3, 1, -1, 3, -1, 3, 2, 1, -1, 2, 3, 2, 3, 2, -1, 2, 1, 0, 1, 2, -1, 3, -1, 2, 2, 1, 2, 1, -1, 2, 2, 1, -1, 3, -1, 3, 4, 0, 1, 1 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

Write positive integers on a 2D grid starting with 1 in the top left corner and continue along the triangular maze as in A056023.

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 (conjectured).

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

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

EXAMPLE

Consider positive integers placed on the plane along a triangular maze:

1 2 6 7 15 16 ...

3 5 8 14 17 ...

4 9 13 18 ...

10 12 19 ...

11 20 ...

21 ...

...

1 is not prime and in 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.

Replacing n by 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 * 2 1 1 * 2 1 1 * ...

* * 4 3 * 3 3 3 * 2 2 2

2 4 * 3 2 * * 2 1 2 * 1

1 3 * 3 2 3 3 2 1 1 1 2

* 3 2 2 * 2 2 * 2 1 . 1

2 * 1 1 3 * 3 2 * 2 1 1

1 2 3 2 3 * 3 2 3 * 1 .

1 2 * * 3 2 2 * 2 1 2 2

* 2 2 4 * 2 1 2 3 2 2 *

1 1 . 2 * 3 1 1 * * 2 3

. 1 2 3 3 * 2 2 3 2 1 1

1 2 * * 2 1 2 * 1 . . 1

...

In order to produce sequence graph is read along original mapping.

MATHEMATICA

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 *)

PROG

(Java) See Links section.

CROSSREFS

Cf. A056023 - plane mapping

Different arrangements of integers:

Cf. A326405 - antidiagonals,

Cf. A326407 - square mapping,

Cf. A326408 - square maze,

Cf. A326409 - Hamiltonian path,

Cf. A326410 - Ulam's spiral.

Sequence in context: A107297 A107296 A080847 * A334006 A270572 A095276

Adjacent sequences: A326403 A326404 A326405 * A326407 A326408 A326409

KEYWORD

sign,tabl

AUTHOR

Witold Tatkiewicz, Oct 02 2019

STATUS

approved

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Last modified November 30 14:55 EST 2022. Contains 358442 sequences. (Running on oeis4.)