%I #21 Aug 12 2018 21:27:53
%S 1,2,6,3,12,4,10,5,20,8,14,16,18,9,15,21,7,28,22,24,26,30,25,35,40,32,
%T 34,36,38,42,27,33,39,45,48,56,44,70,46,50,52,13,65,60,51,75,55,66,54,
%U 57,72,58,62,64,68,78,63,69,81,84,80,74,76,82,41
%N Arrange numbers in a clockwise spiral with initial terms a(1)=1, a(2)=2, a(4)=3, a(6)=4, a(8)=5; thereafter each number shares a factor with each of its four (N,S,E,W) neighbors.
%C Start with smallest number which has not yet appeared and satisfies the conditions: a(3)=6; thereafter always choose smallest number which has not yet appeared and satisfies the conditions.
%C This is a two-dimensional spiral analog of EKG sequence A064413.
%C In A064413 we have initial terms in the positions 1,2.
%C In the two-dimensional case we have 4 sides.
%C So the initial TERMS are
%C 5
%C 4 1 2 (1)
%C 3
%C But the POSITIONS in the spiral are indexed thus:
%C .
%C 7--8--9--10
%C |
%C 6 1--2
%C | |
%C 5--4--3
%C .
%C So the initial terms, by (1), are a(1)=1, a(2)=2, a(4)=3, a(6)=4, a(8)=5.
%C Conjecture: The sequence is a permutation of the positive integers. - _Vladimir Shevelev_, May 06 2015
%H Peter J. C. Moses, <a href="/A253279/b253279.txt">Table of n, a(n) for n = 1..5625</a>
%H Peter J. C. Moses, <a href="/A253279/a253279.pdf">The first few squares.</a>
%e The spiral begins
%e .
%e 26--30--25--35--40--32 etc.
%e |
%e 24 10---5--20---8
%e | | |
%e 22 4 1---2 14
%e | | | |
%e 28 12---3---6 16
%e | |
%e 7--21--15---9--18
%Y Cf. A064413, A257321-A257340, A257112.
%K nonn
%O 1,2
%A _Vladimir Shevelev_, May 02 2015
%E Correction of a(42) and more terms from _Peter J. C. Moses_, May 03 2015