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

1,1

Comments

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

Links

Table of n, a(n) for n=1..66.

Example

   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}

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]

Crossrefs

Cf. A121339, A000037.

Sequence in context: A064664 A352188 A323637 * A078386 A163254 A277696

Adjacent sequences: A349634 A349635 A349636 * A349638 A349639 A349640

Keyword

nonn

Author

Giorgos Kalogeropoulos, Nov 23 2021

Status

approved