A235362 -id:A235362 - cp's OEIS frontend

A270714 Decimal expansion of (1/2)^(1/3).

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

Offset: 0

Clark Kimberling, Mar 22 2016

Let c = (1/2)^(1/3). A sphere of radius c*r has half the volume of a sphere of radius r. - Rick L. Shepherd, Aug 12 2016

Let c = (1/2)^(1/3). The relative maximum of xy(x+y)=1 is (c,-1/c^2). - Clark Kimberling, Oct 05 2020

			0.79370052598409973737585281963615413019574666394992650490414288091260825...

Clark Kimberling, Table of n, a(n) for n = 0..10000

Cf. A002580, A005480, A235362, A269993.

Mathematica
```
RealDigits[(1/2)^(1/3), 10, 200][[1]]
```
PARI
```
(1/2)^(1/3) \\ Altug Alkan, Mar 22 2016
```

Equals 1/A002580 = A002580*A235362 = A005480*A020761. [corrected and expanded by Rick L. Shepherd, Aug 12 2016]

Equals Product_{k>=1} (1 + (-1)^k/(3*k+1)). - Amiram Eldar, Aug 10 2020

A182774 Beatty sequence for 1+2^(-2/3).

Original entry on oeis.org

1, 3, 4, 6, 8, 9, 11, 13, 14, 16, 17, 19, 21, 22, 24, 26, 27, 29, 30, 32, 34, 35, 37, 39, 40, 42, 44, 45, 47, 48, 50, 52, 53, 55, 57, 58, 60, 61, 63, 65, 66, 68, 70, 71, 73, 74, 76, 78, 79, 81, 83, 84, 86, 88, 89, 91, 92, 94, 96, 97, 99, 101, 102, 104

Offset: 1

Clark Kimberling, Nov 30 2010

nonn

Let u=2^(1/3). Jointly rank {ju} and {k/u} as in the first comment at A182760; a(n) is the position of n/u. A182774 is the complement of A182773.

Vincenzo Librandi, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Beatty Sequence.
Index entries for sequences related to Beatty sequences.

Cf. A182760, A182773, A182774, A235362.

[Floor(n*(1+2^(-2/3))): n in [1..80]]; // Vincenzo Librandi, Oct 25 2011

Floor[Range[100]*(1 + 2^(-2/3))] (* Paolo Xausa, Jul 09 2024 *)

a(n) = floor(n*(1 + 2^(-2/3))).

A239797 Decimal expansion of square root of 3 divided by cube root of 4.

Original entry on oeis.org

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

Offset: 1

Alonso del Arte, Mar 27 2014

This is the principal square root of 3 divided by the principal cube root of 4. This number is the imaginary part of a complex cubic root of 2, namely -2^(1/3)/2 + sqrt(-3)/4^(1/3). (The other complex cubic root of 2 is the same except for the sign of the imaginary part.)

An algebraic number of degree 6. - Charles R Greathouse IV, Apr 14 2014

			1.0911236359717214...

G. C. Greubel, Table of n, a(n) for n = 1..10000
William Stein, Algebraic Number Theory, a Computational Approach, p. 69 (in the PDF), Example 6.1.1, or Discriminants and Norms chapter (HTML).
Index entries for algebraic numbers, degree 6

Cf. A235362.

RealDigits[Sqrt[3]/4^(1/3), 10, 100][[1]]

polrootsreal(16*x^6-27)[2] \\ Charles R Greathouse IV, Apr 14 2014

2^(1/3)/2 = 1/2^(2/3) = 1/4^(1/3).
(-2^(1/3)/2 + sqrt(-3)/4^(1/3))^3 = 2.
Equals Product_{n >= 1} 1/(1 - 1/(6*n - 2)^2 ). - Fred Daniel Kline, Dec 19 2015

A355178 Decimal expansion of 2^(-2/3)/L, where L is the conjectured Landau's constant A081760.

Original entry on oeis.org

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

Offset: 1

Stefano Spezia, Sep 23 2022

			1.159595266963928365769992051570020881945...

Markus Faulhuber, An application of hypergeometric functions to heat kernels on rectangular and hexagonal tori and a "Weltkonstante"-or-how Ramanujan split temperatures, The Ramanujan Journal volume 54, pages 1-27 (2021). See p. 23.

Cf. A002580, A073005, A081760, A175379, A203145, A235362, A357318.

First[RealDigits[N[2^(1/3)*Gamma[1/6]/(2Gamma[1/3]Gamma[5/6]), 90]]]

Equals A002580*A357318.
Equals A235362/A081760 = A235362*A175379/(A073005*A203145).
Equals Sum_{k,m in Z^2} exp(-Pi*(2/sqrt(3))*(k^2+k*m+m^2)).
From Gerry Martens, Jul 29 2023: (Start)
Equals hypergeom([1/3, 2/3], [1], 1/2).
Equals sqrt(Pi)/(Gamma(2/3)*Gamma(5/6)). (End)

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A270714 Decimal expansion of (1/2)^(1/3).

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A182774 Beatty sequence for 1+2^(-2/3).

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A239797 Decimal expansion of square root of 3 divided by cube root of 4.

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A355178 Decimal expansion of 2^(-2/3)/L, where L is the conjectured Landau's constant A081760.

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