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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. 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 (list; graph; refs; listen; history; text; internal format)
 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

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Last modified September 19 04:40 EDT 2024. Contains 376004 sequences. (Running on oeis4.)