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
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
KEYWORD
sign,tabl
AUTHOR
Witold Tatkiewicz, Oct 07 2019
STATUS
approved