A092039 -id:A092039 - 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)

A232814 Decimal expansion of the minimum surface index of a half-open cylinder.

Original entry on oeis.org

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

Offset: 1

Stanislav Sykora, Dec 04 2013

Equivalently, the minimum external surface area among all pots or cups (half-open cylinders) delimiting a unit volume. It is attained when the cylinder is right circular and its height matches the radius of its base. We might call it the 'cooking pot constant'. It minimizes the material needed to make the pot (and also its cooling rate). Its determination is a nice exercise in elementary analysis.

For a general definition of surface index, see A232808.

			4.3937756626845697890604275817913711752157905668838115230717678...

Stanislav Sykora, Table of n, a(n) for n = 1..1000
Wikipedia, Cylinder (geometry).

Cf. Other surface indices: A232808 (spheres), A232809, A232810, A232811, A232812 (platonic solids), A232813 (closed cylinders), A232815 (closed cones), A232816 (open cones), A232817 (open tubes).

Cf. A092039.

RealDigits[3 * Surd[Pi, 3], 10, 120][[1]] (* Amiram Eldar, May 17 2023 *)

Equals 3*Pi^(1/3) = 3*A092039.

A093204 Decimal expansion of Pi^(-1/3).

Original entry on oeis.org

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

Offset: 0

Mohammad K. Azarian, Apr 22 2004

			0.682784063255295681467020833...

Index entries for transcendental numbers

Cf. A000796, A002161, A049541, A197111.

RealDigits[Surd[Pi,-3],10,120][[1]] (* Harvey P. Dale, Aug 18 2014 *)

PARI
```
Pi^(-1/3) \\ M. F. Hasler, Oct 07 2014
```

1/A092039. - M. F. Hasler, Oct 07 2014

A240977 Beatty sequence for cube root of Pi: a(n) = floor(n*Pi^(1/3)).

Original entry on oeis.org

0, 1, 2, 4, 5, 7, 8, 10, 11, 13, 14, 16, 17, 19, 20, 21, 23, 24, 26, 27, 29, 30, 32, 33, 35, 36, 38, 39, 41, 42, 43, 45, 46, 48, 49, 51, 52, 54, 55, 57, 58, 60, 61, 62, 64, 65, 67, 68, 70, 71, 73, 74, 76, 77, 79, 80, 82, 83, 84

Offset: 0

Sarah Nathanson, Sep 30 2014

nonn

Beatty complement of A248524. - M. F. Hasler, Oct 07 2014

G. C. Greubel, Table of n, a(n) for n = 0..5000
Index entries for sequences related to Beatty sequences

Cf. A092039 (Pi^(1/3)), A022844 (similar for Pi), A037086 (similar for sqrt(Pi)), A248524.

static int a(int n) {return (int) (n*Math.pow(Math.PI,(1.0/3)));}

Table[Floor[n*(Pi^(1/3))], {n, 0, 50}] (* G. C. Greubel, Feb 14 2017 *)

a(n)=n\Pi^(-1/3) \\ M. F. Hasler, Oct 07 2014

a(n) = floor(n*(Pi^1/3)).

a(0)=0 prepended by Eric M. Schmidt, Oct 06 2014

Name edited by M. F. Hasler, Oct 07 2014

A197111 Continued fraction for cube root of Pi and its inverse.

Original entry on oeis.org

0, 1, 2, 6, 1, 1, 3, 1, 1, 1, 2, 2, 1, 1, 2, 7, 1, 5, 5, 53, 3, 29, 3, 2, 6, 1, 1, 2, 1, 4, 8, 3, 2, 2, 1, 13, 1, 3, 1, 2, 1, 1, 1, 1, 2, 11, 4, 1, 37, 1, 142, 2, 1, 1, 8, 1, 19, 1, 2, 1, 1, 1, 1, 1, 2, 1, 1, 3, 24, 1, 1, 1, 7, 1, 55, 9, 1, 1, 1, 224, 2

Offset: 0

Michael Lee, Oct 10 2011

Starting with a(0) this is the cont.frac. of Pi^(-1/3), and starting with a(1) the cont.frac. of Pi^(1/3). - M. F. Hasler, Oct 07 2014

			Pi^(1/3) = 1.4645918875615232630201425272637903917385968556279371743572...

Cf. A092039, A093204.

Mathematica
```
ContinuedFraction[Pi^(1/3)]
```

contfrac(Pi^(-1/3)) \\ M. F. Hasler, Oct 07 2014

A248524 Beatty sequence for 1/(1-Pi^(-1/3)).

Original entry on oeis.org

3, 6, 9, 12, 15, 18, 22, 25, 28, 31, 34, 37, 40, 44, 47, 50, 53, 56, 59, 63, 66, 69, 72, 75, 78, 81, 85, 88, 91, 94, 97, 100, 104, 107, 110, 113, 116, 119, 122, 126, 129, 132, 135, 138, 141, 145, 148, 151, 154, 157, 160, 163, 167, 170, 173, 176, 179, 182

Offset: 1

M. F. Hasler, Oct 07 2014

nonn

Beatty complement of A240977.

G. C. Greubel, Table of n, a(n) for n = 1..1000

Cf. A092039 (Pi^(1/3)), A093204 (Pi^(-1/3)), A022844 (Beatty seq. for Pi), A037086 (Beatty seq. for sqrt(Pi)).

Table[Floor[n/(1 - Pi^(-1/3))], {n, 1, 50}] (* G. C. Greubel, Apr 06 2017 *)

PARI
```
a(n)=n\(1-Pi^(-1/3))
```

a(n) = floor(n/(1-Pi^(-1/3))).

cp's OEIS Frontend

A002580 Decimal expansion of cube root of 2.

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A232814 Decimal expansion of the minimum surface index of a half-open cylinder.

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A093204 Decimal expansion of Pi^(-1/3).

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A240977 Beatty sequence for cube root of Pi: a(n) = floor(n*Pi^(1/3)).

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A197111 Continued fraction for cube root of Pi and its inverse.

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A248524 Beatty sequence for 1/(1-Pi^(-1/3)).

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