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Lexicographically earliest sequence of positive integers such that the values A345415(a(n), a(n+1)) are all distinct.
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%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