

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(n1) is the empty set.


0



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
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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



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, n1], k]Intersection[pcf@a[n1], pcf@k]!={}, k++]; k); Array[a, 100]


CROSSREFS



KEYWORD

nonn


AUTHOR



STATUS

approved



