This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A346429 #13 Jul 18 2021 11:04:24 %S A346429 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, %T A346429 28,29,30,13,14,32,31,56,55,54,53,86,127,126,85,84,83,82,81,80,79,78, %U A346429 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 %N 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. %C A346429 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. %C A346429 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. %C A346429 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. %H A346429 Scott R. Shannon, <a href="/A346429/a346429.gif">Image of the first 5000 steps</a>. 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. %H A346429 Scott R. Shannon, <a href="/A346429/a346429_1.gif">Image of the first 1000000 steps</a>. 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. %e A346429 The square spiral is numbered as follows: %e A346429 . %e A346429 17--16--15--14--13 . %e A346429 | | . %e A346429 18 5---4---3 12 29 %e A346429 | | | | | %e A346429 19 6 1---2 11 28 %e A346429 | | | | %e A346429 20 7---8---9--10 27 %e A346429 | | %e A346429 21--22--23--24--25--26 %e A346429 . %e A346429 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. %e A346429 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. %Y A346429 Cf. A335661, A330979, A332767, A335364, A344367. %K A346429 nonn %O A346429 1,2 %A A346429 _Scott R. Shannon_, Jul 17 2021