A121339 -id:A121339 - cp's OEIS frontend

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

Giorgos Kalogeropoulos, Nov 23 2021

nonn

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

			   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}

Cf. A121339, A000037.

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]

A345475 Nonsquares k whose continued fraction for the square root of k has a periodic part that is a nondecreasing sequence.

Original entry on oeis.org

2, 3, 5, 6, 7, 8, 10, 11, 12, 13, 15, 17, 18, 20, 24, 26, 27, 30, 32, 35, 37, 38, 39, 40, 41, 42, 48, 50, 51, 55, 56, 58, 63, 65, 66, 68, 72, 74, 75, 80, 82, 83, 84, 87, 90, 99, 101, 102, 104, 105, 110, 120, 122, 123, 130, 132, 135, 136, 143, 145, 146, 147

Offset: 1

Giorgos Kalogeropoulos, Sep 16 2021

nonn

All k = m^2 + 1 (A002522) belong in the sequence because the periodic part of the continued fraction of sqrt(k) has a single element.

			a(5)=7 because the periodic part of the continued fraction of sqrt(7) is (1,1,1,4) which is a nondecreasing sequence.
19 is not a term because the periodic part of the continued fraction of sqrt(19) is (2, 1, 3, 1, 2, 8) which is not a nondecreasing sequence.

Cf. A003285, A002522, A121339.

Select[Range@147,!IntegerQ@Sqrt@#&&OrderedQ@Last@ContinuedFraction[Sqrt@#]&]

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.

Original entry on oeis.org

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

Giorgos Kalogeropoulos, Feb 16 2022

nonn

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.

			   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}

Cf. A121339, A349637.

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]

cp's OEIS Frontend

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.

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A345475 Nonsquares k whose continued fraction for the square root of k has a periodic part that is a nondecreasing sequence.

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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.

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Mathematica