

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
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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, Minesweeperstyle 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



