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.

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, 39, 23, 30, 43, 50, 29, 51, 31, 41, 32, 42, 34, 55, 35, 56, 47, 65, 33, 66, 44, 68, 45, 82, 46, 83, 48, 72, 53, 84, 52, 87, 54, 101, 57, 89, 58, 90, 62, 102, 59, 104

Offset: 1

Author

Giorgos Kalogeropoulos, Nov 23 2021

Keywords

• nonn

Comments

Conjecture: This is a permutation of the nonsquares (A000037).

Examples

			   n  a(n)  Periodic part of continued fraction for square root of a(n)
  --  ----  -----------------------------------------------------------
   1    2   {2}
   2    5   {4}
   3    3   {1, 2}
   4   10   {6}
   5    6   {2, 4}
   6   11   {3, 6}
   7    7   {1, 1, 1, 4}
   8   12   {2, 6}
   9    8   {1, 4}
  10   17   {8}
  11   13   {1, 1, 1, 1, 6}
		

Crossrefs

Cf. A121339, A000037.

Programs

Mathematica

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]