A003814 -id:A003814 - cp's OEIS frontend

A031626 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 38.

3373, 9241, 30010, 30706, 44221, 45065, 61673, 62669, 82021, 82594, 85489, 105914, 107873, 132857, 160441, 192373, 195893, 196778, 198554, 200338, 201233, 227201, 228154, 230066, 232949, 233914, 271129, 305545, 307757, 309977, 314441

Offset: 1

David W. Wilson

nonn

Subsequence of A003814.

First term 1445 removed by Georg Fischer, Jun 16 2019

A031627 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 39.

Original entry on oeis.org

9629, 31057, 31765, 32122, 66733, 86074, 110513, 111178, 112514, 114533, 138745, 140237, 168541, 170185, 173497, 203029, 205738, 209378, 238769, 239746, 241706, 242689, 243674, 244661, 246641, 248629, 249626, 323465, 325741, 328025

Offset: 1

David W. Wilson

nonn

Subsequence of A003814.

First term 1522 removed by Georg Fischer, Jun 16 2019

A031628 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 40.

Original entry on oeis.org

401, 10429, 20498, 32842, 33937, 68813, 90826, 92641, 94474, 178525, 180217, 213973, 214898, 216754, 251626, 252629, 253634, 254641, 256661, 257674, 259706, 262769, 344237, 346585, 348941, 353677, 386602, 392837, 396602, 400385, 439333, 443314, 449989

Offset: 1

David W. Wilson

nonn

Subsequence of A003814.

op40Q[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]] ==40]; Select[Range[500000],op40Q] (* Harvey P. Dale, Dec 06 2015 *)

Corrected and extended by Harvey P. Dale, Dec 06 2015

A031664 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 76.

Original entry on oeis.org

119797, 176521, 250169, 327409, 420193, 422789, 425393, 426698, 530345, 533261, 643645, 653305, 766154, 774929, 780218, 783754, 905026, 906929, 910741, 914561, 918389, 920306, 926069, 1061629, 1219657, 1224077, 1232941, 1241837

Offset: 1

David W. Wilson

nonn

Subsequence of A003814.

First term 5777 removed by Georg Fischer, Jun 16 2019

A031666 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 78.

Original entry on oeis.org

13933, 123985, 124690, 126106, 185021, 258233, 260269, 262313, 347146, 550925, 553897, 559865, 679417, 823178, 953201, 957109, 959066, 962986, 964949, 966914, 972821, 974794, 1120093, 1286797, 1295885, 1467482, 1474757, 1477186

Offset: 1

David W. Wilson

nonn

Subsequence of A003814.

cf78Q[n_]:=Module[{s=Sqrt[n],len,cf},cf=If[IntegerQ[s],{1,1},ContinuedFraction[s][[2]]];len= Length[ cf];OddQ[len]&&cf[[Floor[len/2]]]==cf[[Ceiling[len/2]]]==78]; Select[Range[ 1480000],cf78Q] (* Harvey P. Dale, Dec 16 2023 *)

First term 6085 removed by Georg Fischer, Jun 16 2019

A031668 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 80.

Original entry on oeis.org

40829, 132577, 195485, 197257, 362629, 365041, 466778, 473633, 584057, 587117, 590185, 850613, 852458, 1002626, 1006634, 1014674, 1016689, 1018706, 1024769, 1184473, 1188829, 1351085, 1360397, 1365065, 1374425, 1541042, 1548497

Offset: 1

David W. Wilson

nonn

Subsequence of A003814.

cf80Q[n_]:=Module[{s=Sqrt[n],cf,len},cf=If[IntegerQ[s],{0,0},ContinuedFraction[s][[2]]];len=Length[cf];OddQ[len]&&cf[[Floor[len/2]]]==cf[[Ceiling[len/2]]]==80]; Select[Range[155*10^4],cf80Q] (* Harvey P. Dale, Jul 24 2022 *)

First term 6401 removed by Georg Fischer, Jun 16 2019

A031670 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 82.

Original entry on oeis.org

42461, 136981, 137722, 206237, 208057, 379681, 383386, 387109, 487493, 493093, 497314, 621305, 750397, 897338, 899233, 903029, 1053301, 1055354, 1057409, 1059466, 1065649, 1069781, 1073921, 1075994, 1228393, 1237273, 1250653

Offset: 1

David W. Wilson

nonn

Subsequence of A003814.

First term 6725 removed by Georg Fischer, Jun 16 2019

A031671 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 83.

Original entry on oeis.org

85313, 85898, 142210, 291769, 392101, 394609, 395866, 504389, 630797, 633977, 764317, 771325, 916378, 927898, 1078069, 1086389, 1090561, 1098929, 1277629, 1282153, 1460105, 1469785, 1474637, 1479497, 1662482, 1665061, 1667642

Offset: 1

David W. Wilson

nonn

Subsequence of A003814.

First term 6890 removed by Georg Fischer, Jun 16 2019

A350546 Numbers k such that the period of the continued fraction for sqrt(k) is a prime.

Original entry on oeis.org

3, 6, 8, 11, 12, 13, 15, 18, 20, 24, 27, 29, 30, 35, 38, 39, 40, 41, 42, 48, 51, 53, 56, 58, 61, 63, 66, 68, 72, 73, 74, 80, 83, 84, 85, 87, 89, 90, 97, 99, 102, 104, 105, 110, 120, 123, 125, 130, 132, 143, 146, 147, 148, 150, 152, 156, 157, 168, 171, 173, 182, 185, 193, 195, 198, 200

Offset: 1

Giorgos Kalogeropoulos, Jan 04 2022

nonn

			13 is a term because the continued fraction for sqrt(13) is (3;1,1,1,1,6,1,1,1,1,6,...), whose periodic part is (1,1,1,1,6); its length (the period) is 5 (a prime).

Cf. A003285, A206586, A003814.

Select[Range@200,PrimeQ@Length@Last@ContinuedFraction[Sqrt[#]]&]

isokf(n, p) = {localprec(p); my(cf = contfrac(sqrt(n))); setsearch(Set(cf), 2*cf[1]); }
f(n) = {if (issquare(n), 0, my(p=100); while (! isokf(n, p), p+=100); localprec(p); my(cf = contfrac(sqrt(n))); for (k=2, #cf, if (cf[k] == 2*cf[1], return (k-1))); ); } \\ A003285
isok(k) = isprime(f(k)); \\ Michel Marcus, Jan 05 2022

cp's OEIS Frontend

A031626 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 38.

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A031627 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 39.

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A031628 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 40.

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A031664 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 76.

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A031666 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 78.

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A031668 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 80.

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A031670 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 82.

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A031671 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 83.

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A350546 Numbers k such that the period of the continued fraction for sqrt(k) is a prime.

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