cp's OEIS Frontend

A351652 a(1) = 1; for n > 1, a(n) is the smallest positive integer not occurring earlier such that the intersection of the periodic parts of the continued fractions for square roots of a(n) and a(n-1) is the empty set.

%I A351652 #10 Apr 11 2022 21:55:14
%S A351652 1,2,4,3,5,9,6,10,7,11,8,12,16,13,17,14,18,15,20,25,19,26,21,27,22,36,
%T A351652 23,30,24,28,37,29,38,31,39,32,40,33,49,34,41,35,42,50,43,51,44,64,45,
%U A351652 65,46,66,47,55,48,56,68,53,72,57,81,52,82,54,83,58,84,59
%N A351652 a(1) = 1; for n > 1, a(n) is the smallest positive integer not occurring earlier such that the intersection of the periodic parts of the continued fractions for square roots of a(n) and a(n-1) is the empty set.
%C A351652 Conjecture: This is a permutation of the positive integers.
%C A351652 The conjecture is true: we can always extend the sequence with a square, so eventuality every square will appear; also, after a square, we can always extend the sequence with the least number not yet in the sequence. - _Rémy Sigrist_, Mar 12 2022
%C A351652 The periodic part of the continued fraction for the square root of a square is the empty set.
%e A351652    n  a(n)  Periodic part of continued fraction for square root of a(n)
%e A351652   --  ----  -----------------------------------------------------------
%e A351652    1    1   {}
%e A351652    2    2   {2}
%e A351652    3    4   {}
%e A351652    4    3   {1,2}
%e A351652    5    5   {4}
%e A351652    6    9   {}
%e A351652    7    6   {2, 4}
%e A351652    8   10   {6}
%e A351652    9    7   {1, 1, 1, 4}
%e A351652   10   11   {3, 6}
%e A351652   11    8   {1, 4}
%t A351652 pcf[m_]:=If[IntegerQ[Sqrt@m],{},Last@ContinuedFraction@Sqrt@m];
%t A351652 a[1]=1;a[n_]:=a[n]=(k=2;While[MemberQ[Array[a,n-1],k]||Intersection[pcf@a[n-1],pcf@k]!={},k++];k);Array[a,100]
%Y A351652 Cf. A121339, A349637.
%K A351652 nonn
%O A351652 1,2
%A A351652 _Giorgos Kalogeropoulos_, Feb 16 2022