A005480 -id:A005480 - cp's OEIS frontend

A002947 Continued fraction for cube root of 4.

1, 1, 1, 2, 2, 1, 3, 2, 3, 1, 3, 1, 30, 1, 4, 1, 2, 9, 6, 4, 1, 1, 2, 7, 2, 3, 2, 1, 6, 1, 1, 1, 25, 1, 7, 7, 1, 1, 1, 1, 266, 1, 3, 2, 1, 3, 60, 1, 5, 1, 8, 5, 6, 1, 4, 20, 1, 4, 1, 1, 14, 1, 4, 4, 1, 1, 1, 1, 7, 3, 1, 1, 2, 1, 3, 1, 4, 4, 1, 1, 1, 3, 1, 34, 8, 2, 10, 6, 3, 1, 2, 31, 1, 1, 1, 4, 3, 44, 1, 45

Offset: 0

N. J. A. Sloane

			4^(1/3) = 1.58740105196819947... = 1 + 1/(1 + 1/(1 + 1/(2 + 1/(2 + ...)))). - _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
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]
Herman P. Robinson, Letter to N. J. A. Sloane, Nov 13 1973.
Gang Xiao, Contfrac
Index entries for continued fractions for constants

Cf. A005480 (decimal expansion). - Harry J. Smith, May 08 2009

Cf. A002355, A002356 (convergents).

[ContinuedFraction(4^(1/3))]; // Vincenzo Librandi, Aug 02 2015

ContinuedFraction[4^(1/3), 80] (* Alonso del Arte, Jul 24 2015 *)

{ allocatemem(932245000); default(realprecision, 21000); x=contfrac(4^(1/3)); for (n=1, 20000, write("b002947.txt", n-1, " ", x[n])); } \\ Harry J. Smith, May 08 2009

More terms from Antonio G. Astudillo (afg_astudillo(AT)lycos.com), Mar 29 2003

Offset changed by Andrew Howroyd, Jul 04 2024

A329219 Decimal expansion of 2^(10/12) = 2^(5/6).

Original entry on oeis.org

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

Offset: 1

Jianing Song, Nov 08 2019

2^(10/12) is the ratio of the frequencies of the pitches in a minor seventh (e.g., D4-C5) in 12-tone equal temperament.

			1.78179743...

Wikipedia, Equal temperament
Wikipedia, Twelfth root of two
Index entries for sequences based on music

Frequency ratios of musical intervals:
Perfect unison:   2^(0/12) = 1.0000000000
Minor second:     2^(1/12) = 1.0594630943...  (A010774)
Major second:     2^(2/12) = 1.1224620483...  (A010768)
Minor third:      2^(3/12) = 1.1892071150...  (A010767)
Major third:      2^(4/12) = 1.2599210498...  (A002580)
Perfect fourth:   2^(5/12) = 1.3348398541...  (A329216)
Aug. fourth/
Dim. fifth:       2^(6/12) = 1.4142135623...  (A002193)
Perfect fifth:    2^(7/12) = 1.4983070768...  (A328229)
Minor sixth:      2^(8/12) = 1.5874010519...  (A005480)
Major sixth:      2^(9/12) = 1.6817928305...  (A011006)
Minor seventh:   2^(10/12) = 1.7817974362...  (this sequence)
Major seventh:   2^(11/12) = 1.8877486253...  (A329220)
Perfect octave:  2^(12/12) = 2.0000000000

First[RealDigits[2^(5/6), 10, 100]] (* Paolo Xausa, Apr 27 2024 *)

PARI
```
default(realprecision, 100); 2^(10/12)
```

Equals 2/A010768.

Equals Product_{k>=0} (1 + (-1)^k/(6*k + 1)). - Amiram Eldar, Jul 25 2020

A329216 Decimal expansion of 2^(5/12).

Original entry on oeis.org

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

Offset: 1

Jianing Song, Nov 08 2019

2^(5/12) is the ratio of the frequencies of the pitches in a perfect fourth (e.g., D4-G4) in 12-tone equal temperament.

Wikipedia, Equal temperament
Wikipedia, Twelfth root of two
Index entries for sequences based on music

Frequency ratios of musical intervals:
Perfect unison:   2^(0/12) = 1.0000000000
Minor second:     2^(1/12) = 1.0594630943...  (A010774)
Major second:     2^(2/12) = 1.1224620483...  (A010768)
Minor third:      2^(3/12) = 1.1892071150...  (A010767)
Major third:      2^(4/12) = 1.2599210498...  (A002580)
Perfect fourth:   2^(5/12) = 1.3348398541...  (this sequence)
Aug. fourth/
Dim. fifth:       2^(6/12) = 1.4142135623...  (A002193)
Perfect fifth:    2^(7/12) = 1.4983070768...  (A328229)
Minor sixth:      2^(8/12) = 1.5874010519...  (A005480)
Major sixth:      2^(9/12) = 1.6817928305...  (A011006)
Minor seventh:   2^(10/12) = 1.7817974362...  (A329219)
Major seventh:   2^(11/12) = 1.8877486253...  (A329220)
Perfect octave:  2^(12/12) = 2.0000000000

First[RealDigits[2^(5/12), 10, 100]] (* Paolo Xausa, Apr 28 2024 *)

PARI
```
default(realprecision, 100); 2^(5/12)
```

Equals 2/A328229.

A329220 Decimal expansion of 2^(11/12).

Original entry on oeis.org

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

Offset: 1

Jianing Song, Nov 08 2019

2^(11/12) is the ratio of the frequencies of the pitches in a major seventh (e.g., D4-C#5) in 12-tone equal temperament.

Wikipedia, Equal temperament
Wikipedia, Twelfth root of two
Index entries for sequences based on music

Frequency ratios of musical intervals:
Perfect unison:   2^(0/12) = 1.0000000000
Minor second:     2^(1/12) = 1.0594630943...  (A010774)
Major second:     2^(2/12) = 1.1224620483...  (A010768)
Minor third:      2^(3/12) = 1.1892071150...  (A010767)
Major third:      2^(4/12) = 1.2599210498...  (A002580)
Perfect fourth:   2^(5/12) = 1.3348398541...  (A329216)
Aug. fourth/
Dim. fifth:       2^(6/12) = 1.4142135623...  (A002193)
Perfect fifth:    2^(7/12) = 1.4983070768...  (A328229)
Minor sixth:      2^(8/12) = 1.5874010519...  (A005480)
Major sixth:      2^(9/12) = 1.6817928305...  (A011006)
Minor seventh:   2^(10/12) = 1.7817974362...  (A329219)
Major seventh:   2^(11/12) = 1.8877486253...  (this sequence)
Perfect octave:  2^(12/12) = 2.0000000000

First[RealDigits[2^(11/12), 10, 100]] (* Paolo Xausa, Apr 28 2024 *)

PARI
```
default(realprecision, 100); 2^(11/12)
```

Equals 2/A010774.

Equals Product_{k>=0} (1 + (-1)^k/(12*k + 1)). - Amiram Eldar, Jul 29 2020

A347792 Beatty sequence for 2^(2/3).

Original entry on oeis.org

0, 1, 3, 4, 6, 7, 9, 11, 12, 14, 15, 17, 19, 20, 22, 23, 25, 26, 28, 30, 31, 33, 34, 36, 38, 39, 41, 42, 44, 46, 47, 49, 50, 52, 53, 55, 57, 58, 60, 61, 63, 65, 66, 68, 69, 71, 73, 74, 76, 77, 79, 80, 82, 84, 85, 87, 88, 90, 92, 93, 95, 96, 98, 100, 101, 103

Offset: 0

Clark Kimberling, Oct 31 2021

Index entries for sequences related to Beatty sequences

Cf. A005480, A038129, A347793.

Table[Floor[n 2^(2/3)], {n, 0, 200}]  (* A347792 *)

a(n)=sqrtnint(4*n^3,3) \\ Charles R Greathouse IV, Nov 01 2021

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.

A246722 Decimal expansion of Hermite's constant gamma_7 = 2^(6/7).

Original entry on oeis.org

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

Offset: 1

Jean-François Alcover, Sep 02 2014

Also seventh root of 64. - Alonso del Arte, Feb 07 2015

			1.81144732852781334318834574643020637540089176251587471...

Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 8.7 Hermite's Constants, p. 507.

Chai Wah Wu, Table of n, a(n) for n = 1..10000
Steven R. Finch, Errata and Addenda to Mathematical Constants, arXiv:2001.00578 [math.HO], 2020, p. 62.

Cf. A222071 (Hermite's delta_7), A011149, A005480.

Mathematica
```
RealDigits[2^(6/7), 10, 103] // First
```

sqrtn(64, 7) \\ Michel Marcus, Feb 08 2015

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

A251735 Decimal expansion of Sum_{n>=1} (-1)^(n+1)/n^(1/3).

Original entry on oeis.org

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

Offset: 0

R. J. Mathar, Dec 07 2014

Cubic root analog of A113024.

			0.57175283382527766493...

G. C. Greubel, Table of n, a(n) for n = 0..10000

Cf. A005480, A113024, A251734.

Maple
```
Zeta(1/3)*(1-root[3](4)) ; evalf(%) ;
```

RealDigits[-Zeta[1/3]*(4^(1/3) - 1), 10, 100][[1]] (* G. C. Greubel, Apr 15 2018 *)

-zeta(1/3)*(4^(1/3)-1) \\ Charles R Greathouse IV, Apr 20 2016

Equals 1 - 1/A002580 + 1/A002581 - 1/A005480 + ... = A251734 *(1 - A005480).

A317969 Decimal expansion of (2^(1/3)-1)^(1/3).

Original entry on oeis.org

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

Offset: 0

N. J. A. Sloane, Aug 27 2018

(2^(1/3)-1)^(1/3) = (1/9)^(1/3) - (2/9)^(1/3) + (4/9)^(1/3) is a famous and remarkable identity of Ramanujan's.

Ramanujan's question 1076 (ii), see Berndt and Rankin in References: Show that (4*(2/3)^(1/3)-5*(1/3)^(1/3))^(1/8) = (4/9)^(1/3)-(2/9)^(1/3)+(1/9)^(1/3). - Hugo Pfoertner, Aug 28 2018

			0.638185820860644153015503659444067701265157543977997683421...

B. C. Berndt and R. A. Rankin, Ramanujan: Essays and Surveys, American Mathematical Society, 2001, ISBN 0-8218-2624-7, page 222 (JIMS 11, page 199).
Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, Section 1.1.2, p. 4.
S. Ramanujan, Coll. Papers, Chelsea, 1962, page 331, Question 682; page 334 Question 1076.

Susan Landau, Simplification of nested radicals, SIAM Journal on Computing 21.1 (1992): 85-110. See page 85. [Do not confuse this paper with the short FOCS conference paper with the same title, which is only a few pages long.]
S. Ramanujan, Question 682, Journal of the Indian Mathematical Society, VII, p. 160.
S. Ramanujan, Question 1076, Journal of the Indian Mathematical Society, XI, p. 199.
Vincent Thill, Radicaux et Ramanujan, April 2021, see k.

Cf. A318526.

Cf. A002580, A005480, A108369.

evalf((4*(2/3)^(1/3)-5*(1/3)^(1/3))^(1/8)); # Muniru A Asiru, Aug 28 2018

RealDigits[N[Power[Power[2, (3)^-1] - 1, (3)^-1], 100]] (* Peter Cullen Burbery, Apr 09 2022 *)

(4*(2/3)^(1/3)-5*(1/3)^(1/3))^(1/8) /* Hugo Pfoertner Aug 28 2018 */

sqrtn(1/9, 3) - sqrtn(2/9, 3) + sqrtn(4/9, 3) \\ Michel Marcus, Jan 07 2022

From Michel Marcus, Jan 08 2022: (Start)
Equals (A002580-1)^(1/3).
k^(3*n) = x(n) + A002580*y(n) + A005480*z(n) where k is this constant z(n) = A108369(n-1), y(n) = z(n)+z(n+1), x(n) = y(n)+y(n+1); A002580 and A005480 are the cube root of 2 and 4. (End)
Minimal polynomial: 1 - 3*x^3 - 3*x^6 - x^9. - Stefano Spezia, Oct 15 2024

A383267 Decimal expansion of (4/11)^(1/3).

Original entry on oeis.org

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

Offset: 0

Arkadiusz Wesolowski, Apr 21 2025

In the standard cosmology, the temperature of the free-streaming neutrinos which formed the cosmic neutrino background is (4/11)^(1/3) of the relic photon temperature after the electron-positron annihilation in the early universe (assuming that all electrons and positrons annihilated into photons).

			0.713765855503608170671899991762661247590796547589038066915626752084583...

E. W. Kolb and M. S. Turner, The Early Universe, Addison-Wesley, Redwood City, CA, 1990, p. 503 Appendix A.
R. E. Lopez, S. Dodelson, A. Heckler and M. S. Turner, Precision detection of the cosmic neutrino background, Physical Review Letters 82 (1999) 3952-3955, p. 3952.
Steven Weinberg, Gravitation and Cosmology Principles and Applications of the General Theory of Relativity, John Wiley, New York, 1972, p. 537.

Wikipedia, Cosmic neutrino background.

Cf. A111728.

SetDefaultRealField(RealField(106)); n:=(4/11)^(1/3); Reverse(Intseq(Floor(10^105*n)));

RealDigits[(4/11)^(1/3),10,105][[1]] (* Stefano Spezia, Apr 25 2025 *)

PARI
```
(4/11)^(1/3)
```

Equals 1/A111728 = A005480/A010583.

cp's OEIS Frontend

A002947 Continued fraction for cube root of 4.

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A329219 Decimal expansion of 2^(10/12) = 2^(5/6).

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A329216 Decimal expansion of 2^(5/12).

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A329220 Decimal expansion of 2^(11/12).

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A347792 Beatty sequence for 2^(2/3).

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

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Maple

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A246722 Decimal expansion of Hermite's constant gamma_7 = 2^(6/7).

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