|
| |
|
|
A326407
|
|
Minesweeper sequence of positive integers arranged on a 2D grid along a square array that grows by alternately adding a row at its bottom edge and a column at its right edge.
|
|
8
|
|
|
|
2, -1, -1, 2, -1, 5, -1, 2, 1, 3, -1, 4, -1, 3, 2, 0, -1, 3, -1, 3, 3, 2, -1, 1, 0, 2, 3, 2, -1, 3, -1, 1, 2, 2, 2, 0, -1, 1, 2, 3, -1, 3, -1, 3, 3, 2, -1, 1, 0, 1, 2, 2, -1, 2, 3, 2, 3, 2, -1, 2, -1, 3, 2, 0, 1, 2, -1, 2, 1, 1, -1, 3, -1, 1, 1, 1, 3, 3, -1, 1, 0
(list;
table;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
|
OFFSET
|
1,1
|
|
|
COMMENTS
|
Place positive integers on a 2D grid starting with 1 in the top left corner and continue along an increasing square array as in A060734.
Replace each prime with -1 and each nonprime with the number of primes in adjacent grid cells around them.
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) < 6 (conjectured).
a(n) = 5 for n={6} (conjectured).
Set of n such that a(n) = 4 is unbounded (conjectured).
|
|
|
LINKS
|
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).
|
|
|
EXAMPLE
|
Consider positive integers distributed onto the plane along an increasing square array:
1 4 9 16 25 36
2 3 8 15 24 35
5 6 7 14 23 34
10 11 12 13 22 33
17 18 19 20 21 32
26 27 28 29 30 31
...
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 2 primes: 3, and 7. Therefore a(8) = 2.
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 2 1 . . . . . . . . . ...
* * 2 2 1 2 1 2 1 1 . 1
* 5 * 3 * 2 * 3 * 2 1 1
3 * 4 * 2 2 2 * 3 * 1 1
* 3 * 3 3 1 3 2 3 1 2 1
2 3 2 * 3 * 3 * 1 . 1 *
* 1 2 3 * 3 * 2 1 . 2 3
1 2 2 * 2 3 2 3 1 2 2 *
1 2 * 2 1 1 * 3 * 2 * 2
2 * 3 2 . 2 3 * 3 3 1 1
* 3 * 1 1 2 * 3 * 2 1 .
1 2 1 2 2 * 3 3 2 * 1 1
...
In order to produce the sequence, the graph is read along the original mapping.
|
|
|
MATHEMATICA
|
Block[{n = 12, s}, s = ArrayPad[Array[If[#1 < 2 #2 - 1, #2^2 + #2 - #1, (#1 - #2)^2 + #2] & @@ {#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, Oct 02 2019 *)
|
|
|
PROG
|
(Java) See Links section.
|
|
|
CROSSREFS
|
Different arrangements of integers:
|
|
|
KEYWORD
|
|
|
|
AUTHOR
|
|
|
|
STATUS
|
approved
|
| |
|
|
