A346429 Squares visited on a square spiral when stepping to the closest unvisited square that contains a number with a different number of divisors to the number in the current square. If two or more such squares are the same distance from the current square then the one with the smallest number is chosen.

1, 2, 9, 8, 7, 6, 5, 4, 3, 12, 11, 10, 25, 24, 23, 22, 45, 46, 47, 48, 49, 26, 50, 51, 52, 27, 28, 29, 30, 13, 14, 32, 31, 56, 55, 54, 53, 86, 127, 126, 85, 84, 83, 82, 81, 80, 79, 78, 77, 76, 115, 114, 75, 74, 43, 42, 21, 20, 19, 18, 17, 16, 15, 61, 34, 60, 33, 59, 58, 92, 57, 90, 89, 88, 87

Offset: 1

Author

Scott R. Shannon, Jul 17 2021

Keywords

• nonn

Comments

The first term at which a step to a non-adjacent square is required is a(64) = 61; the previous square 15 having neighbors already visited or with four divisors.

The linked images show that the path of visited squares can approach the origin after many terms. For example 44 is not visited until the 973644th step, although 43 and 45 are visited after 54 and 16 steps respectively. It is possible eventually all squares are visited although this is unknown.

In the first 10 million terms the longest step distance between terms is on the 8836645th step, between 1548859 and 1578754, a distance of ~90.2 units.

Examples

			The square spiral is numbered as follows:
.
  17--16--15--14--13   .
   |               |   .
  18   5---4---3  12   29
   |   |       |   |   |
  19   6   1---2  11   28
   |   |           |   |
  20   7---8---9--10   27
   |                   |
  21--22--23--24--25--26
.
a(3) = 9 as a(2) = 2 which has two divisors, and the closest three unvisited squares around 2 are 3, 11 and 9, and of those only 9 has a divisor count not equal to two.
a(10) = 12 as a(9) = 3 which has two divisors, and the closest two unvisited squares around 3 are 12 and 14. Both have more than two divisors but 12 is the smaller so it the square stepped to.
		

Links

• Scott R. Shannon, Image of the first 5000 steps. The colors are graduated across the spectrum to show the relative step order. Note how after about 4670 step the path approaches the origin again. The central 1 square is marked with a white dot while the smallest unvisited square, 44, is marked with a yellow dot. The longest step distance ~10.6 units after 4681 steps is shown as a white line. Click the image to zoom in.
• Scott R. Shannon, Image of the first 1000000 steps. Note the numerous paths that approach the origin. The smallest unvisited square, 159, is marked with a yellow dot. The longest step distance ~43.8 units after 973504 steps is shown as a while line. Click the image to zoom in.

Crossrefs

Cf. A335661, A330979, A332767, A335364, A344367.