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.

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, 23, 30, 24, 28, 37, 29, 38, 31, 39, 32, 40, 33, 49, 34, 41, 35, 42, 50, 43, 51, 44, 64, 45, 65, 46, 66, 47, 55, 48, 56, 68, 53, 72, 57, 81, 52, 82, 54, 83, 58, 84, 59

Offset

1,2

Comments

Conjecture: This is a permutation of the positive integers.

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

The periodic part of the continued fraction for the square root of a square is the empty set.

Links

Table of n, a(n) for n=1..68.

Example

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

Mathematica

pcf[m_]:=If[IntegerQ[Sqrt@m], {}, Last@ContinuedFraction@Sqrt@m];
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]

Crossrefs

Cf. A121339, A349637.

Sequence in context: A185910 A306779 A349947 * A333087 A091451 A365389

Adjacent sequences: A351649 A351650 A351651 * A351653 A351654 A351655

Keyword

nonn

Author

Giorgos Kalogeropoulos, Feb 16 2022

Status

approved