OFFSET
1,1
COMMENTS
Place positive integers on a 2D grid starting with 1 in the center and continue along a spiral.
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 for n = 12.
Set of n such that a(n) = 4 is unbounded (conjecture).
LINKS
Michael De Vlieger, Table of n, a(n) for n = 1..10201 (51 spiral iterations).
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 10^3 spiral iterations read along original mapping, with black indicating a prime and levels of gray commensurate to a(n).
Wikipedia, Minesweeper game
EXAMPLE
Consider positive integers distributed onto the plane along the square spiral:
.
37--36--35--34--33--32--31
| |
38 17--16--15--14--13 30
| | | |
39 18 5---4---3 12 29
| | | | | |
40 19 6 1---2 11 28
| | | | |
41 20 7---8---9--10 27
| | |
42 21--22--23--24--25--26
|
43--44--45--46--47--48--49--...
.
1 is not prime and in adjacent grid cells there are 4 primes: 2, 3, 5 and 7. Therefore a(1) = 4.
2 is prime, therefore a(2) = -1.
8 is not prime and in adjacent grid cells there are 4 primes: 2, 7 and 23. 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---2---1---3---3---*
| |
3 *---2---2---2---* 3
| | | |
3 3 *---3---* 5 *
| | | | | |
2 * 3 4---* * 3
| | | | |
* 3 *---3---3---2 2
| | |
3 3---2---*---2---1---.
|
*---1---1---2---*---2---1---...
In order to produce the sequence, the graph is read along the square spiral.
CROSSREFS
Cf. A136626 - similar sequence: For every number n in Ulam's spiral the sequence gives the number of primes around it (number n excluded).
Cf. A136627 - similar sequence: For every number n in Ulam's spiral the sequence gives the number of primes around it (number n included).
Different arrangements of integers:
KEYWORD
sign,tabl
AUTHOR
Witold Tatkiewicz, Oct 07 2019
STATUS
approved