%I #9 Jul 02 2021 03:34:48
%S 1,1,2,3,5,7,5,8,11,13,11,18,23,13,15,13,21,19,21,23,18,31,19,30,37,
%T 30,41,39,23,25,34,43,28,43,41,43,45,37,35,37,50,47,38,41,54,61,52,61,
%U 59,57,55,53,64,53,70,53,76,37,69,49,66,73,71,69,71,73,75
%N Lexicographically earliest sequence of positive integers such that the values A345415(a(n), a(n+1)) are all distinct.
%C When writing gcd(a(n), a(n+1)) as u*a(n) + v*a(n+1) where u, v are minimal, the u's are all distinct.
%H Rémy Sigrist, <a href="/A345872/b345872.txt">Table of n, a(n) for n = 1..10000</a>
%H Rémy Sigrist, <a href="/A345872/a345872.gp.txt">PARI program for A345872</a>
%e The table A345415(n, k) begins:
%e n\k| 1 2 3 4
%e ---+--------------
%e 1| 0 1 1 1
%e 2| 0 0 -1 1
%e 2| 0 1 0 -1
%e 2| 0 0 1 0
%e For n = 1:
%e - we can choose a(1) = 1.
%e For n = 2:
%e - we can choose a(2) = 1,
%e - A345415(a(1), a(2)) = 0.
%e For n = 3:
%e - a(3) must be different from 1,
%e - we can choose a(3) = 2,
%e - A345415(a(2), a(3)) = 1.
%e For n = 4:
%e - a(4) must be different from 1 and from 2,
%e - we can choose a(4) = 3.
%o (PARI) See Links section.
%Y Cf. A345415, A345873.
%K nonn
%O 1,3
%A _Rémy Sigrist_, Jun 27 2021