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

1,2

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.

Links

Antoine Mathys, Table of n, a(n) for n = 1..10000

Project Euler, Problem 128 - Hexagonal Tile Differences.

Example

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
.

Prog

(C++)
#include <iostream>
bool is_prime (long n)
{
  if (n < 2) return false;
  if ((n == 2) || (n == 3)) return true;
  if ((n % 6 != 1) && (n % 6 != 5)) return false;
  for (long x = 1; (6 * x - 1) * (6 * x - 1) <= n; x++) {
    if (n % (6 * x - 1) == 0) return false;
    if ((6 * x + 1) * (6 * x + 1) > n) break;
    if (n % (6 * x + 1) == 0) return false;
  }
  return true;
}
int main ()
{
  constexpr long MAX_INDEX = 10000;
  if (MAX_INDEX >= 1) std::cout << "1 1\n";
  if (MAX_INDEX >= 2) std::cout << "2 2\n";
  // We count the rings starting at 0 (the one containing 1)
  for (long r = 2, index = 2; index < MAX_INDEX; r++) {
    if (!is_prime (6 * r - 1)) continue;
    // first cell
    if (!is_prime (6 * r + 1)) goto LAST;
    if (!is_prime (12 * r + 5)) goto LAST;
    index++;
    std::cout << index << " " << 3 * (r - 1) * r + 2 << '\n';
    if (index == MAX_INDEX) break;
LAST:
    // last cell
    if (!is_prime (6 * r + 5)) continue;
    if (!is_prime (12 * (r - 1) + 5)) continue;
    index++;
    std::cout << index << " " << 3 * r * (r + 1) + 1 << '\n';
  }
}

Crossrefs

Sequence in context: A134789 A058217 A306616 * A183183 A072675 A033711

Adjacent sequences: A372220 A372221 A372222 * A372224 A372225 A372226

Keyword

nonn

Author

Antoine Mathys, Apr 22 2024

Status

approved