A005482 -id:A005482 - cp's OEIS frontend

A002580 Decimal expansion of cube root of 2.

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

Offset: 1

N. J. A. Sloane

2^(1/3) is Hermite's constant gamma_3. - Jean-François Alcover, Sep 02 2014, after Steven Finch.
For doubling the cube using origami and a standard geometric construction employing two right angles see the W. Lang link, Application 2, p. 14, and the references given there. See also the L. Newton link. - Wolfdieter Lang, Sep 02 2014
Length of an edge of a cube with volume 2. - Jared Kish, Oct 16 2014
For any positive real c, the mappings R(x)=(c*x)^(1/4) and S(x)=sqrt(c/x) have the same unique attractor c^(1/3), to which their iterated applications converge from any complex plane point. The present case is obtained setting c=2. It is noteworthy that in this way one can evaluate cube roots using only square roots. The CROSSREFS list some other cases of cube roots to which this comment might apply. - Stanislav Sykora, Nov 11 2015
The cube root of any positive number can be connected to the Philo lines (or Philon lines) for a 90-degree angle. If the equation x^3-2 is represented using Lill's method, it can be shown that the path of the root 2^(1/3) creates the shortest segment (Philo line) from the x axis through (1,2) to the y axis. For more details see the article "Lill's method and the Philo Line for Right Angles" linked below. - Raul Prisacariu, Apr 06 2024

			1.2599210498948731647672106072782283505702514...

John H. Conway and Richard K. Guy, The Book of Numbers, New York: Springer-Verlag, 1996. See pp. 192-193.
Jan Gullberg, Mathematics from the Birth of Numbers, W. W. Norton & Co., NY & London, 1997, §3.4 Irrational Numbers and §12.3 Euclidean Construction, pp. 84, 421.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
Horace S. Uhler, Many-figure approximations for cubed root of 2, cubed root of 3, cubed root of 4, and cubed root of 9 with chi2 data. Scripta Math. 18, (1952). 173-176.
David Wells, The Penguin Dictionary of Curious and Interesting Numbers. Penguin Books, NY, 1986, Revised edition 1987, pp. 33-34.

Harry J. Smith, Table of n, a(n) for n = 1..20000
Laszlo C. Bardos, Double a Cube, CutOutFoldUp.
Steven R. Finch, Errata and Addenda to Mathematical Constants, p. 62.
Wolfdieter Lang, Notes on Some Geometric and Algebraic Problems solved by Origami, arXiv:1409.4799 [math.MG], 2014.
Liz Newton, The power of origami.
Raul Prisacariu, Lill's method and the Philo Line for Right Angles.
Simon Plouffe, Plouffe's Inverter, The cube root of 2 to 20000 digits.
Simon Plouffe, 2**(1/3) to 2000 places.
Simon Plouffe, Generalized expansion of real constants.
H. S. Uhler, Many-figure approximations for cubed root of 2, cubed root of 3, cubed root of 4, and cubed root of 9 with chi2 data, Scripta Math. 18, (1952). 173-176. [Annotated scanned copies of pages 175 and 176 only]
Eric Weisstein's World of Mathematics, Delian Constant.
Index entries for algebraic numbers, degree 3.

Cf. A002945 (continued fraction), A270714 (reciprocal), A253583.
Cf. A246644.
Cf. A002581, A005480, A005481, A005482, A005486, A010581, A010582, A092039, A092041, A139340.

Digits:=100: evalf(2^(1/3)); # Wesley Ivan Hurt, Nov 12 2015

RealDigits[N[2^(1/3), 5!]] (* Vladimir Joseph Stephan Orlovsky, Sep 04 2008 *)

default(realprecision, 20080); x=2^(1/3); for (n=1, 20000, d=floor(x); x=(x-d)*10; write("b002580.txt", n, " ", d));  \\ Harry J. Smith, May 07 2009

default(realprecision, 100); x= 2^(1/3); for(n=1, 100, d=floor(x); x=(x-d)*10; print1(d, ", "))  \\ Altug Alkan, Nov 14 2015

(-2^(1/3) - 2^(1/3) * sqrt(-3))^3 = (-2^(1/3) + 2^(1/3) * sqrt(-3))^3 = 16. - Alonso del Arte, Jan 04 2015
Set c=2 in the identities c^(1/3) = sqrt(c/sqrt(c/sqrt(c/...))) = sqrt(sqrt(c*sqrt(sqrt(c*sqrt(sqrt(...)))))). - Stanislav Sykora, Nov 11 2015
Equals Product_{k>=0} (1 + (-1)^k/(3*k + 2)). - Amiram Eldar, Jul 25 2020
From Peter Bala, Mar 01 2022: (Start)
Equals Sum_{n >= 0} (1/(3*n+1) - 1/(3*n-2))*binomial(1/3,n) = (3/2)* hypergeom([-1/3, -2/3], [4/3], -1). Cf. A290570.
Equals 4/3 - 4*Sum_{n >= 1} binomial(1/3,2*n+1)/(6*n-1) = (4/3)*hypergeom ([1/2, -1/6], [3/2], 1).
Equals hypergeom([-2/3, -1/6], [1/2], 1).
Equals hypergeom([2/3, 1/6], [4/3], 1). (End)

A005483 Continued fraction for cube root of 7.

Original entry on oeis.org

1, 1, 10, 2, 16, 2, 1, 4, 2, 1, 21, 1, 3, 5, 1, 2, 1, 1, 2, 11, 5, 1, 3, 1, 2, 27, 4, 1, 282, 8, 1, 2, 1, 1, 3, 1, 3, 2, 6, 4, 1, 2, 1, 5, 1, 1, 2, 1, 1, 1, 3, 2, 8, 1, 2, 2, 4, 5, 1, 1, 36, 1, 1, 1, 1, 2, 1, 2, 31, 2, 1, 1, 7, 1, 1, 1, 1, 6, 7, 6, 5, 7, 1, 6, 1

Offset: 0

N. J. A. Sloane

			7^(1/3) = 1.912931182772389... = 1 + 1/(1 + 1/(10 + 1/(2 + 1/(16 + ...)))). - _Harry J. Smith_, May 08 2009

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Harry J. Smith, Table of n, a(n) for n = 0..19999
S. Lang and H. Trotter, Continued fractions for some algebraic numbers, J. Reine Angew. Math. 255 (1972), 112-134.
S. Lang and H. Trotter, Continued fractions for some algebraic numbers, J. Reine Angew. Math. 255 (1972), 112-134. [Annotated scanned copy]
G. Xiao, Contfrac
Index entries for continued fractions for constants

Cf. A005482 (decimal expansion).

Cf. A005484, A005485 (convergents).

ContinuedFraction[Surd[7,3],70] (* Harvey P. Dale, Oct 01 2013 *)

{ allocatemem(932245000); default(realprecision, 21000); x=contfrac(7^(1/3)); for (n=1, 20000, write("b005483.txt", n-1, " ", x[n])); } \\ Harry J. Smith, May 08 2009

Offset changed by Andrew Howroyd, Jul 05 2024

A005484 Numerators of continued fraction convergents to cube root of 7.

Original entry on oeis.org

1, 2, 21, 44, 725, 1494, 2219, 10370, 22959, 33329, 722868, 756197, 2991459, 15713492, 18704951, 53123394, 71828345, 124951739, 321731823, 3664001792, 18641740783, 22305742575, 85558968508, 107864711083, 301288390674, 8242651259281, 33271893427798, 41514544687079

Offset: 0

N. J. A. Sloane, Herman P. Robinson

D. H. Lehmer, Guide to Tables in the Theory of Numbers. Bulletin No. 105, National Research Council, Washington, DC, 1941, p. 67.
P. Seeling, Verwandlung der irrationalen Groesse ... in einen Kettenbruch, Archiv. Math. Phys., 46 (1866), 80-120.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Vincenzo Librandi, Table of n, a(n) for n = 0..999

Cf. A005482, A005483, A005485 (denominators).

Numerator[Convergents[7^(1/3), 40]] (* Vincenzo Librandi, Sep 08 2013 *)

a(n)= contfracpnqn(contfrac(7^(1/3), n))[1, 1]; \\ Michel Marcus, Sep 07 2013

More terms from Michel Marcus, Sep 07 2013
Inserted a(27) by Vincenzo Librandi, Sep 08 2013
Offset changed by Andrew Howroyd, Jul 05 2024

A005485 Denominators of continued fraction convergents to cube root of 7.

Original entry on oeis.org

1, 1, 11, 23, 379, 781, 1160, 5421, 12002, 17423, 377885, 395308, 1563809, 8214353, 9778162, 27770677, 37548839, 65319516, 168187871, 1915386097, 9745118356, 11660504453, 44726631715, 56387136168, 157500904051, 4308911545545, 17393147086231, 21702058631776

Offset: 0

N. J. A. Sloane

D. H. Lehmer, Guide to Tables in the Theory of Numbers. Bulletin No. 105, National Research Council, Washington, DC, 1941, p. 67.
P. Seeling, Verwandlung der irrationalen Groesse ... in einen Kettenbruch, Archiv. Math. Phys., 46 (1866), 80-120.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Vincenzo Librandi, Table of n, a(n) for n = 0..999

Cf. A005482, A005482, A005484 (numerators).

Denominator[Convergents[7^(1/3), 40]] (* Vincenzo Librandi, Sep 08 2013 *)

a(n)= contfracpnqn(contfrac(7^(1/3), n))[2, 1];  \\ Michel Marcus, Sep 07 2013

More terms from Herman P. Robinson.
More terms from Michel Marcus, Sep 07 2013
Inserted a(27) by Vincenzo Librandi, Sep 08 2013
Offset changed by Andrew Howroyd, Jul 05 2024

A236027 Decimal expansion of 5^(1/2) - 7^(1/3).

Original entry on oeis.org

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

Offset: 0

Jaroslav Krizek, Jan 19 2014

Decimal expansion of minimal value of the function delta(n) = (n+1)^(1/2) - sigma(n)^(1/tau(n)) for n = 4, where delta(n) is called the delta-deviation from primality of the number n (see A236025).

			0.32313679472740059521005682...

Cf. A236020, A236021, A236022, A236023, A236024, A236025, A236026.

Cf. A002163, A005482.

RealDigits[Sqrt[5] - Surd[7, 3], 10, 120][[1]] (* Amiram Eldar, Jun 06 2023 *)

Equals A002163 - A005482.

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A002580 Decimal expansion of cube root of 2.

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A005483 Continued fraction for cube root of 7.

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A005484 Numerators of continued fraction convergents to cube root of 7.

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A005485 Denominators of continued fraction convergents to cube root of 7.

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A236027 Decimal expansion of 5^(1/2) - 7^(1/3).

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