A010534 -id:A010534 - cp's OEIS frontend

A248306 Egyptian fraction representation of sqrt(83) (A010534) using a greedy function.

9, 10, 96, 59128, 60492862652, 4028155696720429656035, 17013291528585219660340839803942618904707681, 456446101271720201686717630050003610800823977708023229701905011777418617118028569791687

Offset: 0

Robert G. Wilson v, Oct 04 2014

nonn

Egyptian fraction representations of the square roots: A006487, A224231, A248235-A248322.

Egyptian fraction representations of the cube roots: A129702, A132480-A132574.

Egyptian[nbr_] := Block[{lst = {IntegerPart[nbr]}, cons = N[ FractionalPart[ nbr], 2^20], denom, iter = 8}, While[ iter > 0, denom = Ceiling[ 1/cons]; AppendTo[ lst, denom]; cons -= 1/denom; iter--]; lst]; Egyptian[ Sqrt[ 83]]

A040073 Continued fraction for sqrt(83).

Original entry on oeis.org

9, 9, 18, 9, 18, 9, 18, 9, 18, 9, 18, 9, 18, 9, 18, 9, 18, 9, 18, 9, 18, 9, 18, 9, 18, 9, 18, 9, 18, 9, 18, 9, 18, 9, 18, 9, 18, 9, 18, 9, 18, 9, 18, 9, 18, 9, 18, 9, 18, 9, 18, 9, 18, 9, 18, 9, 18, 9, 18, 9, 18, 9, 18, 9, 18, 9

Offset: 0

N. J. A. Sloane

			9.1104335791442988819456261... = 9 + 1/(9 + 1/(18 + 1/(9 + 1/(18 + ...)))). - _Harry J. Smith_, Jun 10 2009

Harry J. Smith, Table of n, a(n) for n = 0..20000
G. Xiao, Contfrac
Index entries for continued fractions for constants
Index entries for linear recurrences with constant coefficients, signature (0, 1).

Cf. A010534 Decimal expansion. - Harry J. Smith, Jun 10 2009

Digits := 100: convert(evalf(sqrt(N)),confrac,90,'cvgts'):

ContinuedFraction[Sqrt[83],300] (* Vladimir Joseph Stephan Orlovsky, Mar 09 2011 *)
LinearRecurrence[{0,1},{9,9,18},70] (* or *) PadRight[{9},70,{18,9}] (* Harvey P. Dale, Jan 25 2018 *)

{ allocatemem(932245000); default(realprecision, 45000); x=contfrac(sqrt(83)); for (n=0, 20000, write("b040073.txt", n, " ", x[n+1])); } \\ Harry J. Smith, Jun 10 2009

A041146 Numerators of continued fraction convergents to sqrt(83).

Original entry on oeis.org

9, 82, 1485, 13447, 243531, 2205226, 39937599, 361643617, 6549522705, 59307347962, 1074081786021, 9726043422151, 176142863384739, 1595011813884802, 28886355513311175, 261572211433685377, 4737186161319647961, 42896247663310517026, 776869644100908954429

Offset: 0

N. J. A. Sloane

Vincenzo Librandi, Table of n, a(n) for n = 0..70
Index entries for linear recurrences with constant coefficients, signature (0,164,0,-1).

Cf. A010534, A041147.

Numerator[Convergents[Sqrt[83], 30]] (* Vincenzo Librandi, Oct 29 2013 *)
LinearRecurrence[{0,164,0,-1},{9,82,1485,13447},20] (* Harvey P. Dale, Apr 12 2022 *)

G.f.: -(x^3-9*x^2-82*x-9) / (x^4-164*x^2+1). - Colin Barker, Nov 05 2013

More terms from Colin Barker, Nov 05 2013

A041147 Denominators of continued fraction convergents to sqrt(83).

Original entry on oeis.org

1, 9, 163, 1476, 26731, 242055, 4383721, 39695544, 718903513, 6509827161, 117895792411, 1067571958860, 19334191051891, 175075291425879, 3170689436717713, 28711280221885296, 519973733430653041, 4708474881097762665, 85272521593190381011, 772161169219811191764

Offset: 0

N. J. A. Sloane

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Index entries for linear recurrences with constant coefficients, signature (0,164,0,-1).

Cf. A041146, A040073, A020840, A010534.

I:=[1, 9, 163, 1476]; [n le 4 select I[n] else 164*Self(n-2)-Self(n-4): n in [1..30]]; // Vincenzo Librandi, Dec 11 2013

Denominator/@Convergents[Sqrt[83], 50] (* Vladimir Joseph Stephan Orlovsky, Jul 05 2011 *)
CoefficientList[Series[(1 + 9 x - x^2)/(x^4 - 164 x^2 + 1), {x, 0, 30}], x] (* Vincenzo Librandi, Dec 11 2013 *)
LinearRecurrence[{0,164,0,-1},{1,9,163,1476},30] (* Harvey P. Dale, Nov 09 2017 *)

G.f.: -(x^2-9*x-1) / (x^4-164*x^2+1). - Colin Barker, Nov 13 2013

a(n) = 164*a(n-2) - a(n-4). - Vincenzo Librandi, Dec 11 2013

A379556 Decimal expansion of the square root of 5312.

Original entry on oeis.org

7, 2, 8, 8, 3, 4, 6, 8, 6, 3, 3, 1, 5, 4, 3, 9, 1, 0, 5, 5, 5, 6, 5, 0, 0, 8, 8, 3, 7, 5, 0, 9, 3, 5, 3, 5, 2, 0, 7, 9, 3, 1, 1, 3, 3, 4, 6, 1, 1, 9, 6, 4, 6, 8, 1, 9, 9, 7, 5, 5, 0, 7, 7, 2, 0, 5, 2, 8, 1, 6, 9, 5, 5, 4, 2, 6, 7, 2, 4, 5, 3, 2, 4, 6, 8, 2, 7, 3, 1, 6, 8, 7, 4, 3, 3, 4, 1, 7, 9, 7, 3, 9, 1, 3, 9, 7, 5, 4, 1, 0, 7, 0, 0, 6, 4, 5, 5, 9, 7, 0, 2, 1, 7, 5, 1, 0, 6, 6

Offset: 2

Alonso del Arte, Dec 25 2024

Continued fraction begins 72 then 1, 7, 1, 1, 2, 1, 1, 2, 1, 35, 1, 2, 1, 1, 2, 1, 1, 7, 1, 144, repeated.

The number sqrt(5312) is an algebraic integer in Z[sqrt(83)] as sqrt(5312) = 8 sqrt(83). Z[sqrt(83)] is a unique factorization domain, and 5312 factorizes as (9 - sqrt(83))^6 (9 + sqrt(83))^6 sqrt(83)^2.

			72.883468633154391...

The Late Show with Stephen Colbert, "How Does TV Work?" - Stephen Colbert Answers Real Questions From Real Kids. Relevant part starts at 3:10.

Cf. A010534, the square root of 83.

SetDefaultRealField(RealField(100)); Sqrt(5312); // Vincenzo Librandi, Jan 02 2025

Mathematica
```
RealDigits[Sqrt[5312], 10, 100][[1]]
```

import java.math.{BigDecimal, MathContext, RoundingMode}
val num = BigDecimal.valueOf(5312)
val mc = new MathContext(128, RoundingMode.HALF_EVEN)
val root = num.sqrt(mc)
root.toString.replace(".", "").toCharArray.toList

cp's OEIS Frontend

A248306 Egyptian fraction representation of sqrt(83) (A010534) using a greedy function.

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A040073 Continued fraction for sqrt(83).

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A041146 Numerators of continued fraction convergents to sqrt(83).

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A041147 Denominators of continued fraction convergents to sqrt(83).

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Magma

Mathematica

Formula

A379556 Decimal expansion of the square root of 5312.

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Magma

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Scala