A005486 -id:A005486 - 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)

A002949 Continued fraction for cube root of 6.

Original entry on oeis.org

1, 1, 4, 2, 7, 3, 508, 1, 5, 5, 1, 1, 1, 2, 1, 1, 24, 1, 1, 1, 3, 3, 30, 4, 10, 158, 6, 1, 1, 2, 12, 1, 10, 1, 1, 3, 2, 1, 1, 89, 1, 1, 2, 1, 1, 1, 3, 1, 2, 1, 7, 1, 2, 18, 1, 17, 2, 2, 10, 14, 3, 1, 2, 1, 2, 1, 5, 1, 1, 2, 26, 1, 4, 65, 1, 1, 1, 27, 1, 2, 1, 4

Offset: 0

N. J. A. Sloane

			6^(1/3) = 1.81712059283213965... = 1 + 1/(1 + 1/(4 + 1/(2 + 1/(7 + ...)))). - _Harry J. Smith_, May 08 2009

H. P. Robinson, Letter to N. J. A. Sloane, Nov 13 1973.
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
Herman P. Robinson, Letter to N. J. A. Sloane, Nov 13 1973.
J. Shallit & N. J. A. Sloane, Correspondence 1974-1975
G. Xiao, Contfrac
Index entries for continued fractions for constants

Cf. A005486 (decimal expansion).

Cf. A002359, A002360 (convergents).

SetDefaultRealField(RealField(100)); ContinuedFraction(6^(1/3)); // G. C. Greubel, Nov 02 2018

with(numtheory):
cfrac(6^(1/3),100,'quotients'); # Muniru A Asiru, Nov 02 2018

ContinuedFraction[6^(1/3), 100] (* G. C. Greubel, Nov 02 2018 *)

{ allocatemem(932245000); default(realprecision, 21000); x=contfrac(6^(1/3)); for (n=1, 20000, write("b002949.txt", n-1, " ", x[n])); } \\ Harry J. Smith, May 08 2009

Offset changed by Andrew Howroyd, Jul 05 2024

A002359 Denominators of continued fraction convergents to cube root of 6.

Original entry on oeis.org

1, 1, 5, 11, 82, 257, 130638, 130895, 785113, 4056460, 4841573, 8898033, 13739606, 36377245, 50116851, 86494096, 2125975155, 2212469251, 4338444406, 6550913657, 23991185377, 78524469788, 2379725279017, 9597425585856, 98353981137577

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, 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).

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

Cf. A002360 (numerators), A002949, A005486.

Denominator[Convergents[Power[6, (3)^-1],30]] (* Harvey P. Dale, Nov 26 2011 *)

a(n) = contfracpnqn(contfrac(6^(1/3), n))[2, 1]; \\ Michel Marcus, Aug 23 2013

More terms from Herman P. Robinson
More terms from Pab Ter (pabrlos(AT)yahoo.com), May 08 2004
Definition clarified by Harvey P. Dale, Nov 26 2011
Offset changed by Andrew Howroyd, Jul 05 2024

A002360 Numerators of continued fraction convergents to cube root of 6.

Original entry on oeis.org

1, 2, 9, 20, 149, 467, 237385, 237852, 1426645, 7371077, 8797722, 16168799, 24966521, 66101841, 91068362, 157170203, 3863153234, 4020323437, 7883476671, 11903800108, 43594876995, 142688431093, 4324247809785, 17439679670233, 178721044512115, 28255364712584403

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, 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).

Harvey P. Dale, Table of n, a(n) for n = 0..999

Cf. A002359 (denominators), A002949, A005486.

Numerator[Convergents[Power[6, (3)^-1],30]] (* Harvey P. Dale, Oct 16 2011 *)

a(n) = contfracpnqn(contfrac(6^(1/3), n))[1, 1]; \\ Michel Marcus, Aug 23 2013

Definition clarified by, and more terms from, Harvey P. Dale, Oct 16 2011

Offset changed by Andrew Howroyd, Jul 05 2024

A319034 Decimal expansion of the height that minimizes the total surface area of the four triangular faces of a square pyramid of unit volume.

Original entry on oeis.org

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

Offset: 1

Jon E. Schoenfield, Oct 22 2018

A square pyramid with a height of h and a base of size s X s has volume V = (1/3)*s^2*h, so a square pyramid of unit volume has s = sqrt(3/h), and the slant height of each of the four triangular faces is t = sqrt(h^2 + (s/2)^2) = sqrt(h^2 + 3/(4*h)), and the total area of the four faces is A = 4*(s*t/2) = sqrt(12*h^3 + 9)/h; this area is minimized at h = (3/2)^(1/3), where it reaches A = 3^(7/6)*2^(1/3).

If the total surface area of all five faces including the square base is to be minimized, then the resulting height is 6^(1/3) (cf. A005486). - Jon E. Schoenfield, Nov 11 2018

			1.14471424255333186780804221193967700891590692078793...

Eric Weisstein's World of Mathematics, Pyramid.

Cf. A002580, A002581, A005486, A010584.

RealDigits[Surd[3/2, 3], 10, 120][[1]] (* Amiram Eldar, Jun 21 2023 *)

sqrtn(3/2, 3) \\ Michel Marcus, Oct 23 2018

Equals (3/2)^(1/3) = (1/2)*A010584.

Equals A002581/A002580. - Michel Marcus, Oct 23 2018

A010648 Decimal expansion of cube root of 78.

Original entry on oeis.org

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

Offset: 1

N. J. A. Sloane

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

RealDigits[N[78^(1/3), 100]][[1]] (* Vincenzo Librandi, Mar 29 2013 *)

sqrtn(78,3) \\ Charles R Greathouse IV, Apr 14 2014

Equals A005486*A010585 = A002581*A010598. [Bruno Berselli, Mar 29 2013]

A210973 Decimal expansion of cube root of (3/4).

Original entry on oeis.org

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

Offset: 0

Omar E. Pol, Aug 09 2012

Radius of a sphere with volume Pi.

			0.908560296416069829445605878... =  A002581 / A005480.

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

Cube root of A152627.

Cf. A005486.

root[3](3/4) ;evalf(%) ; # R. J. Mathar, Sep 14 2012

RealDigits[Surd[3/4,3],10,120][[1]] (* Harvey P. Dale, Sep 27 2015 *)

sqrtn(3/4,3) \\ Charles R Greathouse IV, Apr 14 2014

(3/4)^(1/3).

A005486/2.

A281143 Decimal expansion of 10!^(1/10).

Original entry on oeis.org

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

Offset: 1

Robert G. Wilson v, Jan 15 2017

Base b such that log_b 10! = 10.

Inspired by the idea of utilizing the log scaled to 10! being 10, i.e., log_b 10! = 10, therefore b = 2^(4/5)*3^(2/5)*5^(1/5)*7^(1/10).

			4.52872868811676476220330933719550879349863167608939046288611476046926255...

Cf. A002193, A005486, A005534, A011020, A011092, A011094, A011101.

RealDigits[2^(4/5) 3^(2/5) 5^(1/5) 7^(1/10), 10, 111][[1]] (* or *)
RealDigits[Solve[Log[b, 10!] == 10, b][[1, 1, 2]], 10, 105][[1]]

Equals A011101 * A011094 * A005534 * A011092.

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A002580 Decimal expansion of cube root of 2.

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A002949 Continued fraction for cube root of 6.

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A002359 Denominators of continued fraction convergents to cube root of 6.

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A002360 Numerators of continued fraction convergents to cube root of 6.

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A319034 Decimal expansion of the height that minimizes the total surface area of the four triangular faces of a square pyramid of unit volume.

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A010648 Decimal expansion of cube root of 78.

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A210973 Decimal expansion of cube root of (3/4).

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