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 A257112 #80 Aug 12 2018 21:28:00 %S A257112 1,2,11,4,55,6,25,8,165,14,3,16,15,26,5,12,35,18,7,22,21,32,9,28,27, %T A257112 10,33,20,77,34,49,38,231,46,121,24,143,36,65,44,45,52,51,58,75,56,39, %U A257112 40,57,50,63,62,69,64,81,68,87,17,93,136,105,74,85,42,95,48,115,54,161 %N A257112 Arrange numbers in a clockwise spiral with initial terms a(1)=1, a(2)=2, a(4)=4, a(6)=6, a(8)=8, a(11)=3, a(15)=5, a(19)=7, a(23)=9; thereafter each number is relatively prime to all of its four (N,S,E,W) neighbors, but shares a factor with each of its (N,S,E,W) neighbors at distance 2 and also satisfies an additional condition stated in the comments. %C A257112 To formulate the additional condition, let us call two numbers strictly connected if the set of prime divisors of one of them is a subset of the set of prime divisors of the other. Then the positions of two strictly connected terms should not be a knight's move apart. %C A257112 Start with smallest number which has not yet appeared and satisfies the conditions: a(3)=11; thereafter always choose smallest number which has not yet appeared and satisfies the conditions. %C A257112 This is a two-dimensional spiral analog of A098550. %C A257112 In A098550 we have initial terms in the positions 1,2,3. %C A257112 In the two-dimensional case we have 4 sides. So the initial TERMS are %C A257112 9 %C A257112 8 %C A257112 7 6 1 2 3 (1) %C A257112 4 %C A257112 5 %C A257112 But the POSITIONS in the spiral are indexed thus: %C A257112 . %C A257112 7--8--9--10 %C A257112 | %C A257112 6 1--2 %C A257112 | | %C A257112 5--4--3 %C A257112 . %C A257112 So the initial terms, by (1), are a(1)=1, a(2)=2, a(4)=4, a(6)=6, a(8)=8, ... %C A257112 Conjecture: the sequence is a permutation of the positive integers. - _Vladimir Shevelev_, May 06 2015 %H A257112 Peter J. C. Moses, <a href="/A257112/b257112.txt">Table of n, a(n) for n = 1..5625</a> %H A257112 Peter J. C. Moses, <a href="/A257112/a257112.pdf">The first few squares.</a> %e A257112 The spiral begins %e A257112 . %e A257112 21---32----9---28---27---10 etc. %e A257112 | %e A257112 22 25----8--165---14 %e A257112 | | | %e A257112 7 6 1----2 3 %e A257112 | | | | %e A257112 18 55----4---11 16 %e A257112 | | %e A257112 35---12----5---26---15 %e A257112 . %e A257112 Formally the smallest a(12) is 10, but then 10 and 5 are strictly connected numbers on a knight move (and a(13) would not exist). So the smallest suitable a(12)=16. %Y A257112 Cf. A098550, A257321-A257340, A255370. %K A257112 nonn %O A257112 1,2 %A A257112 _Vladimir Shevelev_, Apr 24 2015 %E A257112 More terms from _Peter J. C. Moses_, Apr 29 2015