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A349637 a(1) = 2 and a(n) is the smallest nonsquare positive integer not occurring earlier such that the intersection of the periodic parts of continued fractions for square roots of a(n) and a(n-1) is the empty set.

%I A349637 #11 Nov 27 2021 04:53:38
%S A349637 2,5,3,10,6,11,7,12,8,17,13,18,14,26,15,20,27,19,37,21,38,22,40,24,28,
%T A349637 39,23,30,43,50,29,51,31,41,32,42,34,55,35,56,47,65,33,66,44,68,45,82,
%U A349637 46,83,48,72,53,84,52,87,54,101,57,89,58,90,62,102,59,104
%N A349637 a(1) = 2 and a(n) is the smallest nonsquare positive integer not occurring earlier such that the intersection of the periodic parts of continued fractions for square roots of a(n) and a(n-1) is the empty set.
%C A349637 Conjecture: This is a permutation of the nonsquares (A000037).
%e A349637    n  a(n)  Periodic part of continued fraction for square root of a(n)
%e A349637   --  ----  -----------------------------------------------------------
%e A349637    1    2   {2}
%e A349637    2    5   {4}
%e A349637    3    3   {1, 2}
%e A349637    4   10   {6}
%e A349637    5    6   {2, 4}
%e A349637    6   11   {3, 6}
%e A349637    7    7   {1, 1, 1, 4}
%e A349637    8   12   {2, 6}
%e A349637    9    8   {1, 4}
%e A349637   10   17   {8}
%e A349637   11   13   {1, 1, 1, 1, 6}
%t A349637 pcf=Last@*ContinuedFraction@*Sqrt; a[1]=2; a[n_]:=a[n]=(k=2; While[MemberQ[Array[a,n-1],k]||IntegerQ@Sqrt@k||Intersection[pcf@a[n-1],pcf@k]!={},k++];k); Array[a,100]
%Y A349637 Cf. A121339, A000037.
%K A349637 nonn
%O A349637 1,1
%A A349637 _Giorgos Kalogeropoulos_, Nov 23 2021