cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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A031419 Numbers k such that the continued fraction for sqrt(k) has odd period and if the last term of the periodic part is deleted then there are a pair of central terms both equal to 6.

Original entry on oeis.org

109, 281, 865, 922, 1277, 1613, 1769, 1933, 2161, 2341, 2789, 3098, 3653, 3961, 4285, 4457, 5065, 5153, 5713, 5858, 5954, 6101, 6458, 6554, 6709, 7129, 7349, 7681, 8237, 8941, 9242, 9305, 9677, 10177, 10498, 10565, 10693, 10762, 11162, 11365, 11698
Offset: 1

Views

Author

David W. Wilson

Keywords

Links

Crossrefs

Cf. A031404-A031423.
Subsequence of A003814.

Programs

  • Mathematica
    n = 1; t = {}; While[Length[t] < 50, n++; If[! IntegerQ[Sqrt[n]], c = ContinuedFraction[Sqrt[n]]; len = Length[c[[2]]]; If[OddQ[len] && c[[2, (len + 1)/2]] == 6 && c[[2, (len + 1)/2 - 1]] == 6, AppendTo[t, n]]]]; t (* T. D. Noe, Apr 04 2014; corrected by Georg Fischer, Jun 23 2019 *)

Extensions

a(1) corrected by T. D. Noe, Apr 04 2014
a(1) = 10 removed by Georg Fischer, Jun 23 2019

A031420 Numbers k such that the continued fraction for sqrt(k) has odd period and if the last term of the periodic part is deleted then there are a pair of central terms both equal to 7.

Original entry on oeis.org

349, 778, 1105, 1237, 1306, 1565, 1721, 2473, 3361, 3706, 3889, 4133, 4985, 5261, 5545, 6217, 6841, 6929, 7165, 7253, 7418, 7754, 8021, 8273, 8369, 8629, 9089, 9274, 9461, 10034, 10229, 10333, 10729, 11245, 11657, 12077, 12842, 12941, 13385, 13730, 14314
Offset: 1

Views

Author

David W. Wilson

Keywords

Links

Crossrefs

Cf. A031404-A031423.
Subsequence of A003814.

Programs

  • Mathematica
    n = 1; t = {}; While[Length[t] < 50, n++; If[! IntegerQ[Sqrt[n]], c = ContinuedFraction[Sqrt[n]]; len = Length[c[[2]]]; If[OddQ[len] && c[[2, (len + 1)/2]] == 7, AppendTo[t, n]]]]; t (* T. D. Noe, Apr 04 2014 *)
cf7Q[n_]:=Module[{s=Sqrt[n],cf,len},cf=If[IntegerQ[s],{0}, ContinuedFraction[ s] [[2]]];len=Length[cf];OddQ[len]&&Count[Take[cf,{(len+1)/2-1,(len+1)/2+1}],7]>1]; Select[Range[15000],cf7Q]//Quiet (* Harvey P. Dale, Sep 14 2016 *)

Extensions

Initial erroneous term 50 removed by T. D. Noe, Apr 04 2014

A031421 Numbers k such that the continued fraction for sqrt(k) has odd period and if the last term of the periodic part is deleted then there are a pair of central terms both equal to 8.

Original entry on oeis.org

509, 1450, 2237, 2425, 3946, 4778, 5189, 5473, 5618, 5914, 6445, 6761, 7417, 8185, 9178, 9938, 10133, 10426, 10529, 10826, 10933, 11441, 11861, 12074, 12289, 12506, 12829, 13273, 14653, 14765, 15241, 16217, 16586, 16837, 17090, 17989, 18385, 18650, 18778
Offset: 1

Views

Author

David W. Wilson

Keywords

Links

Crossrefs

Cf. A031404-A031423.
Subsequence of A003814.

Programs

  • Mathematica
    n = 1; t = {}; While[Length[t] < 50, n++; If[! IntegerQ[Sqrt[n]], c = ContinuedFraction[Sqrt[n]]; len = Length[c[[2]]]; If[OddQ[len] && c[[2, (len + 1)/2]] == 8 && c[[2, (len + 1)/2 - 1]] == 8, AppendTo[t, n]]]]; t (* T. D. Noe, Apr 04 2014; adapted by Georg Fischer, Jun 23 2019 *)

Extensions

a(1) corrected by T. D. Noe, Apr 04 2014
Data adapted to new definition by Georg Fischer, Jun 23 2019

A031601 Numbers k such that the continued fraction for sqrt(k) has odd period and if the last term of the periodic part is deleted the two central terms are both 13.

Original entry on oeis.org

409, 1181, 2258, 3562, 3802, 3925, 5305, 5897, 8269, 9829, 10834, 11041, 12389, 15737, 16745, 17785, 21757, 23029, 23633, 24554, 25178, 26869, 27194, 27521, 28181, 28514, 28849, 30554, 30901, 32413, 33853, 36073, 36185, 40841, 41362, 41765, 42170
Offset: 1

Views

Author

David W. Wilson

Keywords

Links

Crossrefs

Subsequence of A003814.

Programs

  • Mathematica
    cf13Q[n_]:=Module[{per=ContinuedFraction[Sqrt[n]][[2]]},OddQ[Length[ per]]&&per[[Floor[Length[per]/2]+1]]==13]; nn=43000;With[{terms = Complement[Range[nn], Range[Floor[Sqrt[nn]]]^2]}, Select[terms, cf13Q]] (* Harvey P. Dale, Nov 23 2011 *)

Extensions

Corrected by Harvey P. Dale, Nov 23 2011

A240950 Numbers k such that the continued fraction for sqrt(k) has odd period, omitting those values of k of the form m^2+1.

Original entry on oeis.org

13, 29, 41, 53, 58, 61, 73, 74, 85, 89, 97, 106, 109, 113, 125, 130, 137, 149, 157, 173, 181, 185, 193, 202, 218, 229, 233, 241, 250, 265, 269, 274, 277, 281, 293, 298, 313, 314, 317, 337, 338, 346, 349, 353, 365, 370, 373, 389, 394, 397, 409, 421, 425, 433
Offset: 1

Views

Author

Takao Ito, Aug 04 2014

Keywords

Comments

p^2 - n*q^2 = -1 is solvable for integers p and q.
A031396 is the union of this sequence with A002522.

Examples

			sqrt(13) = [3;1,1,1,1,6].
sqrt(29) = [5;2,1,1,2,10].

Links

Crossrefs

Cf. A002522, A003814, A031396.

Extensions

More terms from Colin Barker, Dec 18 2014

A031603 Numbers k such that the continued fraction for sqrt(k) has odd period and if the last term of the periodic part is deleted the two central terms are both 15.

Original entry on oeis.org

2858, 3074, 5122, 7177, 7517, 9773, 10169, 13681, 14149, 16673, 17189, 17713, 17978, 18514, 20525, 21097, 22265, 22861, 26041, 27337, 27997, 30113, 32213, 33653, 35594, 35969, 37489, 37874, 38261, 39041, 39434, 39829, 45673, 52825, 54785, 55717
Offset: 1

Views

Author

David W. Wilson

Keywords

Crossrefs

Subsequence of A003814.

Programs

  • Mathematica
    opct15Q[n_]:=Module[{sr=Sqrt[n],cf,len},cf=If[IntegerQ[sr],{}, ContinuedFraction[ sr][[2]]];len=Length[cf];OddQ[len]&&Take[cf,{Floor[ len/2],Floor[len/2]+1}]=={15,15}]; Select[Range[56000],opct15Q] (* Harvey P. Dale, Jun 11 2013 *)

Extensions

Corrected by Harvey P. Dale, Jun 11 2013

A031605 Numbers k such that the continued fraction for sqrt(k) has odd period and if the last term of the periodic part is deleted the two central terms are both 17.

Original entry on oeis.org

6010, 6322, 6481, 9337, 12713, 16609, 17386, 18181, 21898, 22193, 22789, 23393, 26605, 27917, 28585, 32125, 34297, 39733, 40529, 41333, 41738, 42554, 45569, 45994, 46421, 48149, 49466, 49909, 53629, 57373, 59293, 61357, 65357, 66377, 70130, 70657
Offset: 1

Views

Author

David W. Wilson

Keywords

Crossrefs

Subsequence of A003814.

Extensions

First term 290 removed by Georg Fischer, Jun 16 2019

A031606 Numbers k such that the continued fraction for sqrt(k) has odd period and if the last term of the periodic part is deleted the two central terms are both 18.

Original entry on oeis.org

82, 2141, 4274, 6970, 7306, 10525, 10937, 14569, 15053, 18994, 20389, 24314, 25253, 25889, 30637, 31337, 36541, 43793, 47618, 51701, 52154, 52609, 53066, 54449, 55381, 62233, 70537, 71597, 72665, 80485, 81050, 82757, 83905, 85061, 85642, 86225, 86810
Offset: 1

Views

Author

David W. Wilson

Keywords

Crossrefs

Subsequence of A003814.

Programs

  • Mathematica
    opct18Q[n_]:=Module[{s=Sqrt[n],cf,len},cf=If[IntegerQ[s],{0,0},ContinuedFraction[s][[2]]];len=Length[cf];OddQ[len] && cf[[(len+1)/2]] == 18]; Select[Range[90000],opct18Q] (* Harvey P. Dale, Nov 12 2013 *)

Extensions

Corrected by Harvey P. Dale, Nov 12 2013

A031607 Numbers k such that the continued fraction for sqrt(k) has odd period and if the last term of the periodic part is deleted the two central terms are both 19.

Original entry on oeis.org

4538, 4673, 7477, 7825, 11785, 16553, 22129, 22426, 26533, 26858, 27514, 28513, 34217, 40441, 43705, 48053, 50258, 52058, 57269, 57746, 58706, 59674, 60161, 61141, 62129, 62626, 68329, 72553, 77017, 78125, 79241, 80365, 81497, 83785, 88577, 90362
Offset: 1

Views

Author

David W. Wilson

Keywords

Crossrefs

Subsequence of A003814.

Extensions

First term 362 removed by Georg Fischer, Jun 16 2019

A031608 Numbers k such that the continued fraction for sqrt(k) has odd period and if the last term of the periodic part is deleted the two central terms are both 20.

Original entry on oeis.org

2729, 5233, 8362, 8917, 24874, 29873, 31973, 37997, 38777, 46237, 54818, 56698, 57173, 58129, 64129, 64634, 65141, 66161, 66674, 67189, 67706, 68746, 74713, 87277, 88457, 89645, 90841, 100186, 101450, 104002, 104645, 115333, 116689, 126541
Offset: 1

Views

Author

David W. Wilson

Keywords

Crossrefs

Subsequence of A003814.

Programs

  • Mathematica
    cf20Q[n_]:=Module[{s=Sqrt[n],cf,len},cf=If[IntegerQ[s],{1,1},ContinuedFraction[ s][[2]]];len=Length[cf];OddQ[len]&&cf[[Floor[len/2]]] == cf[[Ceiling[len/2]]] ==20]; Select[ Range[ 130000],cf20Q] (* Harvey P. Dale, Sep 18 2021 *)

Extensions

First term 401 removed by Georg Fischer, Jun 16 2019
Previous Showing 21-30 of 49 results. Next

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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