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A253279 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.

%I A253279 #21 Aug 12 2018 21:27:53
%S A253279 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 A253279 34,36,38,42,27,33,39,45,48,56,44,70,46,50,52,13,65,60,51,75,55,66,54,
%U A253279 57,72,58,62,64,68,78,63,69,81,84,80,74,76,82,41
%N A253279 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 A253279 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 A253279 This is a two-dimensional spiral analog of EKG sequence A064413.
%C A253279 In A064413 we have initial terms in the positions 1,2.
%C A253279 In the two-dimensional case we have 4 sides.
%C A253279 So the initial TERMS are
%C A253279      5
%C A253279    4 1 2      (1)
%C A253279      3
%C A253279 But the POSITIONS in the spiral are indexed thus:
%C A253279 .
%C A253279    7--8--9--10
%C A253279    |
%C A253279    6  1--2
%C A253279    |     |
%C A253279    5--4--3
%C A253279 .
%C A253279 So the initial terms, by (1), are a(1)=1, a(2)=2, a(4)=3, a(6)=4, a(8)=5.
%C A253279 Conjecture: The sequence is a permutation of the positive integers. - _Vladimir Shevelev_, May 06 2015
%H A253279 Peter J. C. Moses, <a href="/A253279/b253279.txt">Table of n, a(n) for n = 1..5625</a>
%H A253279 Peter J. C. Moses, <a href="/A253279/a253279.pdf">The first few squares.</a>
%e A253279 The spiral begins
%e A253279 .
%e A253279    26--30--25--35--40--32  etc.
%e A253279     |
%e A253279    24  10---5--20---8
%e A253279     |   |           |
%e A253279    22   4   1---2  14
%e A253279     |   |       |   |
%e A253279    28  12---3---6  16
%e A253279     |               |
%e A253279     7--21--15---9--18
%Y A253279 Cf. A064413, A257321-A257340, A257112.
%K A253279 nonn
%O A253279 1,2
%A A253279 _Vladimir Shevelev_, May 02 2015
%E A253279 Correction of a(42) and more terms from _Peter J. C. Moses_, May 03 2015