A031424 -id:A031424 - cp's OEIS frontend

A031749 Numbers k such that the least term in the periodic part of the continued fraction for sqrt(k) is 71.

5043, 20168, 45375, 80664, 126035, 181488, 247023, 322640, 408339, 504120, 609983, 725928, 851955, 988064, 1134255, 1290528, 1456883, 1633320, 1819839, 2016440, 2223123, 2439888, 2666735, 2903664, 3150675, 3407768, 3674943, 3952200, 4239539

Offset: 1

David W. Wilson

nonn

The continued fraction expansion of sqrt((j*m)^2+t*m) for m >= 1 where t divides 2*j has the form [j*m, 2*j/t, 2*j*m, 2*j/t, 2*j*m, ...]. Thus numbers of the form (71*m)^2 + 2*m for m >= 1 are in the sequence. Are there any others? - Chai Wah Wu, Jun 18 2016

The term 25776072 is not of the form (71*m)^2 + 2*m. - Chai Wah Wu, Jun 19 2016

Chai Wah Wu, Table of n, a(n) for n = 1..10000

Cf. A031424.

lt71Q[n_]:=Module[{s=Sqrt[n]},If[IntegerQ[s],0,Min[ContinuedFraction[s] [[2]]]] == 71]; Select[Range[43*10^5],lt71Q] (* Harvey P. Dale, Apr 11 2017 *)

from sympy import continued_fraction_periodic
A031749_list = [n for n, d in ((n, continued_fraction_periodic(0,1,n)[-1]) for n in range(1,10**5)) if isinstance(d, list) and min(d) == 71] # Chai Wah Wu, Jun 09 2017

A276689 Least term in the periodic part of the continued fraction expansion of sqrt(n) or 0 if n is square.

Original entry on oeis.org

0, 0, 2, 1, 0, 4, 2, 1, 1, 0, 6, 3, 2, 1, 1, 1, 0, 8, 4, 1, 2, 1, 1, 1, 1, 0, 10, 5, 2, 1, 2, 1, 1, 1, 1, 1, 0, 12, 6, 4, 3, 2, 2, 1, 1, 1, 1, 1, 1, 0, 14, 7, 1, 1, 1, 2, 2, 1, 1, 1, 1, 1, 1, 1, 0, 16, 8, 1, 4, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 0, 18, 9, 6, 1

Offset: 0

Chai Wah Wu, Sep 28 2016

nonn

If r > 0 is even, then a((rm/2)^2+m) = r for all m >= 1 and a((r^2-2)^2/4 + (r+1)^3) = r.

If r is odd, then a((rm)^2+2m) = r for all m >= 1 and a(r^4 + r^3 + 5(r+1)^2/4) = r.

Chai Wah Wu, Table of n, a(n) for n = 0..10000

Cf. A031424, A003285, A091453, A096491.

from sympy import continued_fraction_periodic
def A276689(n):
    x = continued_fraction_periodic(0,1,n)
    return min(x[1]) if len(x) > 1 else 0

A031428 Numbers k such that the least term in the periodic part of the continued fraction for sqrt(k) is 4.

Original entry on oeis.org

5, 18, 39, 68, 105, 150, 174, 203, 264, 333, 370, 410, 495, 588, 689, 793, 798, 855, 915, 1040, 1105, 1173, 1313, 1314, 1378, 1387, 1462, 1463, 1620, 1785, 1958, 2138, 2139, 2222, 2232, 2328, 2525, 2730, 2834, 2943, 3164, 3393, 3630, 3741

Offset: 1

David W. Wilson

nonn

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000 (first 700 terms from Vincenzo Librandi)

Different from A007742.

Cf. A031424.

Select[Range[4000], !IntegerQ[Sqrt[#]] && Min[ContinuedFraction[Sqrt[#]][[2]]] == 4&] (* Vincenzo Librandi, Feb 06 2012 *)

A031689 Numbers k such that the least term in the periodic part of the continued fraction for sqrt(k) is 11.

Original entry on oeis.org

123, 488, 1095, 1944, 3035, 4368, 5943, 7760, 9819, 12120, 14663, 16152, 17448, 19344, 20475, 23744, 27255, 31008, 35003, 37284, 39240, 43719, 48440, 53403, 53866, 58608, 59093, 64055, 64562, 67128, 69744, 75675, 81848, 88263, 94920, 101819

Offset: 1

David W. Wilson

nonn

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Cf. A031424.

Select[Range[110000],!IntegerQ[Sqrt[#]]&&Min[ContinuedFraction[Sqrt[#]][[2]]]==11&] (* Vincenzo Librandi, Jan 27 2012 *)

Content that was based on the incorrect assumption that a(n)=121*n^2+2*n removed by R. J. Mathar, Nov 18 2010

A031700 Numbers k such that the least term in the periodic part of the continued fraction for sqrt(k) is 22.

Original entry on oeis.org

122, 486, 1092, 1940, 3030, 4362, 5936, 7752, 9810, 12110, 14652, 17436, 20462, 23730, 27240, 30992, 34986, 39222, 43700, 48420, 53382, 58586, 64032, 69720, 70248, 75650, 81822, 88236, 94892, 101790, 108930, 116312, 123936, 131802, 139910

Offset: 1

David W. Wilson

nonn

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Cf. A031424.

Select[Range[200000],!IntegerQ[Sqrt[#]]&&Min[ContinuedFraction[Sqrt[#]][[2]]]==22&] (* Vincenzo Librandi, Feb 06 2012 *)

A031702 Numbers k such that the least term in the periodic part of the continued fraction for sqrt(k) is 24.

Original entry on oeis.org

145, 578, 1299, 2308, 3605, 5190, 7063, 9224, 11673, 14410, 17435, 20748, 24349, 28238, 32415, 36880, 41633, 46674, 52003, 57620, 63525, 69718, 76199, 82968, 90025, 97370, 97994, 105003, 112924, 121133, 129630, 138415, 147488, 156849, 166498

Offset: 1

David W. Wilson

nonn

			The continued fraction for sqrt(97994) is 313, [25, 24, 25, 626], where the smallest term of the periodic part is 24, so 97994 belongs to the sequence.

Chai Wah Wu, Table of n, a(n) for n = 1..10000

Cf. A031424.

Select[Range[200000], !IntegerQ[Sqrt[#]] && Min[ContinuedFraction[Sqrt[#]][[2]]] == 24&] (* Vincenzo Librandi, Feb 06 2012 *)

from sympy import continued_fraction_periodic
A031702_list = [n for n, s in ((i, continued_fraction_periodic(0,1,i)[-1]) for i in range(1,10**5)) if isinstance(s, list) and min(s) == 24] # Chai Wah Wu, Jun 08 2017

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A031749 Numbers k such that the least term in the periodic part of the continued fraction for sqrt(k) is 71.

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A276689 Least term in the periodic part of the continued fraction expansion of sqrt(n) or 0 if n is square.

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A031428 Numbers k such that the least term in the periodic part of the continued fraction for sqrt(k) is 4.

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A031689 Numbers k such that the least term in the periodic part of the continued fraction for sqrt(k) is 11.

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A031700 Numbers k such that the least term in the periodic part of the continued fraction for sqrt(k) is 22.

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A031702 Numbers k such that the least term in the periodic part of the continued fraction for sqrt(k) is 24.

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