A028832 -id:A028832 - cp's OEIS frontend

A374232 Indices of records in A028832.

1, 2, 3, 14, 19, 46, 94, 151, 211, 379, 526, 694, 919, 1324, 1759, 2011, 2326, 3691, 4174, 5086, 6451, 7606, 8254, 10294, 10651, 13126, 17599, 18979, 19231, 21319, 30319, 31606, 32971, 34654, 42379, 46006, 48799, 58774, 76651, 78094, 82471, 85999, 90931, 101599

Offset: 1

Amiram Eldar, Jul 01 2024

nonn

Amiram Eldar, Table of n, a(n) for n = 1..100

Cf. A013645, A028832.

s[n_] := If[IntegerQ@ Sqrt[n], 0, Length @ DeleteDuplicates[ContinuedFraction[Sqrt[n]][[2]]]];
seq[kmax_] := Module[{smax = -1, s1, sq = {}}, Do[If[(s1 = s[k]) > smax, smax = s1; AppendTo[sq, k]], {k, 1, kmax}]; sq]; seq[10^5]

A096492 Number of distinct terms in continued fraction period of square root of n.

Original entry on oeis.org

1, 1, 2, 1, 1, 2, 2, 2, 1, 1, 2, 2, 2, 3, 2, 1, 1, 2, 4, 2, 3, 4, 3, 2, 1, 1, 2, 3, 3, 2, 4, 2, 3, 3, 2, 1, 1, 2, 2, 2, 2, 2, 4, 3, 3, 5, 3, 2, 1, 1, 2, 4, 3, 4, 2, 2, 3, 2, 4, 3, 5, 3, 2, 1, 1, 2, 5, 2, 4, 3, 4, 2, 3, 2, 2, 5, 4, 3, 3, 2, 1, 1, 2, 2, 3, 4, 2, 3, 3, 2, 3, 4, 4, 6, 3, 3, 3, 3, 2, 1, 1, 2, 5, 2, 2

Offset: 1

Labos Elemer, Jun 29 2004

nonn

Essentially the same as A028832. - Amiram Eldar, Nov 10 2021

			n=127: the period={3,1,2,2,7,11,7,2,2,1,3,22},distinct-terms={1,2,3,7,11,22}, so a[127]=6;

Cf. A003285, A013646, A028832, A096491, A096493.

{tc=Table[0, {m}], u=1}; Do[s=Length[Union[Last[ContinuedFraction[n^(1/2)]]]]; tc[[u]]=s;u=u+1, {n, 1, m}], tc

a(n) = 1 if n is a square and a(n) = A028832(n) otherwise. - Amiram Eldar, Nov 10 2021

A096495 Number of distinct terms in the periodic part of the continued fraction for sqrt(prime(n)).

Original entry on oeis.org

1, 2, 1, 2, 2, 2, 1, 4, 3, 3, 4, 1, 2, 4, 3, 3, 4, 5, 5, 4, 3, 3, 2, 3, 3, 1, 5, 4, 6, 3, 6, 4, 3, 6, 5, 7, 5, 6, 3, 3, 6, 6, 6, 5, 1, 7, 8, 3, 2, 3, 3, 6, 5, 5, 1, 4, 2, 7, 7, 5, 6, 3, 6, 6, 6, 5, 8, 6, 5, 4, 4, 3, 7, 3, 9, 4, 3, 7, 1, 6, 6, 8, 7, 6, 3, 2, 5, 7, 5, 9, 4, 6, 9, 8, 4, 4, 6, 6, 8, 9, 8, 2, 4, 6, 10

Offset: 1

Labos Elemer, Jun 29 2004

nonn

			n = 31: prime(31) = 127, and the periodic part is {3,1,2,2,7,11,7,2,2,1,3,22}, so a(31) = 6.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A003285, A028832, A054269, A005980, A096491, A096492, A096493, A096494, A096496.

{te=Table[0, {m}], u=1}; Do[s=Length[Union[Last[ContinuedFraction[Prime[n]^(1/2)]]]]; te[[u]]=s;u=u+1, {n, 1, m}];te
Table[Length[Union[ContinuedFraction[Sqrt[Prime[n]]][[2]]]],{n,110}] (* Harvey P. Dale, Jun 22 2017 *)

a(n) = A028832(A000040(n)). - Amiram Eldar, Nov 10 2021

A028833 Index of redundancy in period of continued fraction for sqrt(n).

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 3, 1, 0, 0, 0, 0, 2, 0, 3, 2, 1, 0, 0, 0, 0, 1, 2, 0, 4, 2, 1, 1, 0, 0, 0, 0, 0, 0, 1, 0, 6, 5, 3, 7, 1, 0, 0, 0, 0, 2, 2, 2, 2, 0, 3, 5, 2, 1, 6, 1, 0, 0, 0, 0, 5, 0, 4, 3, 4, 0, 4, 3, 2, 7, 2, 1, 1, 0, 0, 0, 0, 0, 2, 6, 0, 3, 2, 0, 5, 4, 6, 10, 1, 1, 8, 1, 0, 0, 0, 0

Offset: 1

Olivier Gérard

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A028832, A003285.

a[n_] := If[IntegerQ[Sqrt[n]], 0, Length[p = ContinuedFraction[Sqrt[n]] // Last] - Length[p // Union]]; Table[a[n], {n, 1, 102}] (* Jean-François Alcover, Dec 26 2012 *)

a(n) = A003285(n) - A028832(n).

A374234 Number k such that the periods of the continued fractions of sqrt(k) and sqrt(k+1) have the same distinct terms.

Original entry on oeis.org

7, 41, 44, 55, 74, 112, 135, 137, 207, 218, 275, 279, 314, 335, 389, 474, 611, 818, 874, 884, 986, 1007, 1009, 1129, 1313, 1325, 1462, 1465, 1824, 2330, 2831, 3201, 3502, 3575, 4927, 5520, 6204, 6623, 8150, 8945, 10989, 11627, 11834, 13033, 13727, 13775, 13888

Offset: 1

Amiram Eldar, Jul 01 2024

nonn

			7 is a term since the period of the continued fraction of sqrt(7) is {1, 1, 1, 4} and the period of the continued fraction of sqrt(8) is {1, 4}. The set of distinct terms of both is {1, 4}.
44 is a term since the period of the continued fraction of sqrt(44) is {1, 1, 1, 2, 1, 1, 1, 12} and the period of the continued fraction of sqrt(45) is {1, 2, 2, 2, 1, 12}. The set of distinct terms of both is {1, 2, 12}.

Amiram Eldar, Table of n, a(n) for n = 1..216

Cf. A003285, A028832.

s[n_] := s[n] = If[IntegerQ@ Sqrt[n], 0, Union[ContinuedFraction[Sqrt[n]][[2]]]]; Select[Range[14000], s[#] == s[# + 1] &]

cp's OEIS Frontend

A374232 Indices of records in A028832.

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A096492 Number of distinct terms in continued fraction period of square root of n.

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A096495 Number of distinct terms in the periodic part of the continued fraction for sqrt(prime(n)).

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A028833 Index of redundancy in period of continued fraction for sqrt(n).

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A374234 Number k such that the periods of the continued fractions of sqrt(k) and sqrt(k+1) have the same distinct terms.

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