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%I #8 Jun 11 2021 18:37:50
%S 243,256,256,344,426,516,876,3886,30420,58852
%N Numbers that return to the origin when performing a non-backtracking walk on a 2D square lattice where at each step the walk moves as close as possible to the origin and the step lengths are the ordered prime factors of the integers.
%C This sequence uses the same walk rules as in A344467 but the first step of length 1 is removed, thus the first step is of length 2, the prime factor of 2. See that sequence for further details of the walk's rules.
%C Assuming another term exists it is at least 2x10^9.
%H Scott R. Shannon, <a href="/A343483/a343483.gif">Image of the path up to factors of 243</a>. Each step is shown as a dot, and the path between steps as a line. The colors are graduated across the spectrum from red to violet to show the relative step ordering. The origin is marked with a white dot.
%e 243 is the first term. It takes a surprisingly larger number of steps, 629, before the walk returns to the origin for the first time. 243 is 3^5 and after the factors of 242 the walk is at coordinate (3,-6) relative to the origin, which can be reached with three steps of length 3, the first three factors of 243.
%e 256 is both the second and third term. The second return to the origin is forty steps after the first visit by the factors of 243, and the third is only four steps after the second - this is due to 256 being 2^8. So after the second return to the origin the walk can go back to it immediately with four steps of length 2, walking out the sides of a square.
%Y Cf. A344467, A027746, A332939, A343153, A309500, A000040.
%K nonn,more,walk
%O 1,1
%A _Scott R. Shannon_, Apr 16 2021
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