A372223 Numbers in a hexagonal tiling (seen as concentric rings) which have exactly three neighbors whose difference from it is prime.

1, 2, 8, 19, 20, 37, 61, 128, 217, 271, 398, 919, 1519, 1520, 2978, 3170, 4220, 4447, 4681, 5677, 5941, 6488, 8269, 9920, 10621, 12481, 16651, 17558, 22448, 26227, 29701, 34028, 34669, 35317, 35971, 56719, 60920, 61777, 74419, 75367, 80197, 88238, 93458

Offset: 1

Author

Antoine Mathys, Apr 22 2024

Keywords

• nonn

Comments

A hexagonal tile with number 1 is surrounded by a ring of six hexagonal tiles, starting on the right and numbering the tiles 2 to 7 in a counterclockwise direction. New rings are added in the same fashion, with the next rings being numbered 8 to 19, 20 to 37, and so on (see the illustration below). By finding the difference between tile n and each of its six neighbors we shall define PD(n) to be the number of those differences which are prime. For example, working counter-clockwise around tile 8 the differences are 12, 13, 1, 6, 11, and 29. So PD(8)=3. In the same way, the differences around tile 17 are 1, 10, 11, 1, 16, and 17, hence PD(17) = 2. It can be shown that the maximum value of PD(n) is 3. This sequence lists all tiles for which PD(n)=3 in ascending order.

It can be shown that only the first and last tile of every ring need to be considered.

Examples

			For tile 1, the differences are 1,2,3,4,5,6, thus PD(1)=3 and a(1)=1.
For tile 2, the differences are 6,7,1,1,5,17, thus PD(2)=3 and a(2)=2.
For tile 3, the differences are 6,7,8,1,2,1, thus PD(3)=2 and 3 is not in the list.
Similarly, we see that PD(4)=2, PD(5)=0, PD(6)=2, PD(7)=2, and PD(8)=3. Thus the next term is a(3)=8.
        26--25--24--23
       /             \
      27  12--11--10  22
     /   /         \   \
   28  13   4---3   9  21
   /   /   /     \   \   \
  29  14   5  1  2    8  20
   \   \   \     /    /   /
   30  15   6---7    19 37
    \    \          /   /
     31   16--17--18  36
      \               /
       32---33--34--35
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Links

• Antoine Mathys, Table of n, a(n) for n = 1..10000
• Project Euler, Problem 128 - Hexagonal Tile Differences.