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A348022 The numbers visited on a square spiral when stepping to the smallest unvisited number that is visible from and shares a divisor > 1 with the current number. Start with 1 and 2.

%I A348022 #9 Oct 06 2021 12:48:43
%S A348022 1,2,4,6,3,12,9,15,5,10,14,7,21,27,18,16,8,22,11,33,30,20,24,32,26,13,
%T A348022 39,36,28,35,25,40,44,38,19,76,34,17,68,42,45,51,48,57,66,55,60,46,23,
%U A348022 92,58,50,62,31,155,70,49,56,63,72,64,52,65,78,54,69,84,75,85,80,94,47,188
%N A348022 The numbers visited on a square spiral when stepping to the smallest unvisited number that is visible from and shares a divisor > 1 with the current number. Start with 1 and 2.
%C A348022 A number is visible from the current number if, given it has coordinates (x,y) relative to the current number, the greatest common divisor of |x| and |y| equals 1. See A331400 for the points visible from the starting 1 number.
%C A348022 In the first 10000 terms the longest single step is one at n = 9942 of length sqrt(22570) units between 31002 to 10258. The maximum difference between terms in the same range is from 5171 to 36197 at n = 9977.
%H A348022 Scott R. Shannon, <a href="/A348022/a348022.gif">Image of the path for the first 10000 terms</a>. The colors are graduated across the spectrum to show the relative step order.
%e A348022 The square spiral is numbered as follows:
%e A348022 .
%e A348022   17--16--15--14--13   .
%e A348022    |               |   .
%e A348022   18   5---4---3  12   29
%e A348022    |   |       |   |   |
%e A348022   19   6   1---2  11   28
%e A348022    |   |           |   |
%e A348022   20   7---8---9--10   27
%e A348022    |                   |
%e A348022   21--22--23--24--25--26
%e A348022 .
%e A348022 a(3) = 4 as gcd(4,2) = 2 and 4 is unvisited and visible from 2.
%e A348022 a(4) = 6 as gcd(4,6) = 2 and 6 is unvisited and visible from 4.
%e A348022 a(5) = 3 as gcd(3,6) = 3 and 3 is unvisited and visible from 6.
%e A348022 a(6) = 12 as gcd(12,3) = 3 and 12 is unvisited and visible from 3. Note although 9 is unvisited and gcd(9,3) = 3 it is not visible from 3 due to 2.
%Y A348022 Cf. A348025 (not visible), A331400, A335661, A063826, A332767, A347358.
%K A348022 nonn
%O A348022 1,2
%A A348022 _Scott R. Shannon_, Sep 25 2021