A010582 -id:A010582 - 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)

A010769 Decimal expansion of 7th root of 2.

Original entry on oeis.org

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

Offset: 1

N. J. A. Sloane

This is also the unique positive attractor of the mapping M(x) = sqrt(sqrt(sqrt(2*x))). In general, (p^N-1)-th root of a number f can be approximated by iterating the mapping M(x) = (f*x)^(1/p^N). The convergence is very fast. In this case, p=2, N=3, and f=2. In the form "evaluate the 3rd (or 7th or 15th) root of a number using only square roots", the insight is usable as a recreational math puzzle. - Stanislav Sykora, Oct 26 2015

			1.104089513673812337649505387623...

Vincenzo Librandi, Table of n, a(n) for n = 1..5000

Cube roots (p=2,N=2) for various f: A002580 (2), A002581 (3), A005480 (4), A010582 (10), A092041 (e). 7th roots (p=2,N=3): A246709 (3), A011186 (4), A011201 (5), A011276 (10), A092516 (e). 8th roots (p=3,N=2): A010770 (2), A246710 (3), A011202 (5), A011277 (10). 15th roots (p=2,N=4): A010777(2), A011194(4), A011209(5), A011284(10). - Stanislav Sykora, Oct 26 2015

RealDigits[N[2^(1/7), 100]][[1]] (* Vincenzo Librandi, Apr 02 2013 *)
RealDigits[Surd[2,7],10,120][[1]] (* Harvey P. Dale, Sep 05 2022 *)

sqrtn(2,7) \\ Charles R Greathouse IV, Apr 15 2014

{ default(realprecision, 100); x= 2^(1/7); for(n=1, 100, d=floor(x); x=(x-d)*10; print1(d, ", ")) } \\ Altug Alkan, Nov 14 2015

Equals Product_{k>=0} (1 + (-1)^k/(7*k + 6)). - Amiram Eldar, Jul 29 2020

A010670 Decimal expansion of cube root of 100.

Original entry on oeis.org

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

Offset: 1

N. J. A. Sloane

Heron (or Hero) of Alexandria calculated this constant as 4 + 9/14 in the first century AD, see Deslauriers & Dubuc or Metrica book III section 20. - Charles R Greathouse IV, Jan 12 2012

One "hectoliter" is a non-SI metric unit of volume equal to 100 liters, which is the volume of a cube with an edge of 10*100^(1/3) cm (46.4158883... cm). - Jean-François Alcover, Dec 14 2024

			4.6415888336127788924100763509194465765513491250112436376506928586847778...

Harry J. Smith, Table of n, a(n) for n = 1..20000
G. Deslauriers and S. Dubuc, Le calcul de la racine cubique selon Héron, Elem. Math. 51 (1996), pp. 28-34.

Cf. A010328 (continued fraction), A010582.

RealDigits[N[100^(1/3),200]][[1]] (* Vladimir Joseph Stephan Orlovsky, Jan 24 2012 *)

{ default(realprecision, 20080); x=100^(1/3); for (n=1, 20000, d=floor(x); x=(x-d)*10; write("b010670.txt", n, " ", d)); } \\ Harry J. Smith, May 08 2009

Equals A010582^2. - Hugo Pfoertner, Dec 14 2024

Final digits of sequence corrected using the b-file by N. J. A. Sloane, Aug 30 2009

A010240 Continued fraction for cube root of 10.

Original entry on oeis.org

2, 6, 2, 9, 1, 1, 2, 4, 1, 12, 1, 1, 1, 1, 57, 4, 2, 16, 1, 1, 1, 1, 9, 6, 2, 3, 1, 1, 12, 1, 4, 6, 2, 2, 1001, 3, 2, 6, 9, 1, 15, 1, 2, 1, 1, 27, 2, 1, 1, 21, 1, 11, 9, 2, 18, 15, 2, 25, 1, 1, 1, 1, 35, 30, 2, 4, 3, 10, 1, 3, 3, 4, 1, 1

Offset: 0

N. J. A. Sloane

			10^(1/3) = 2.1544346900318837... = 2 + 1/(6 + 1/(2 + 1/(9 + 1/(1 + ...)))). - _Harry J. Smith_, May 08 2009

Harry J. Smith, Table of n, a(n) for n = 0..20000
G. Xiao, Contfrac
Index entries for continued fractions for constants

Cf. A010582 = Decimal expansion. - Harry J. Smith, May 08 2009

{ allocatemem(932245000); default(realprecision, 21000); x=contfrac(10^(1/3)); for (n=1, 20001, write("b010240.txt", n-1, " ", x[n])); } \\ Harry J. Smith, May 08 2009

cp's OEIS Frontend

A002580 Decimal expansion of cube root of 2.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Maple

Mathematica

PARI

PARI

Formula

A010769 Decimal expansion of 7th root of 2.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

PARI

Formula

A010670 Decimal expansion of cube root of 100.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

Extensions

A010240 Continued fraction for cube root of 10.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

PARI

A010650 Decimal expansion of cube root of 80.

Views

Author

Keywords

Links

Programs

Mathematica

PARI

Formula