A002580 -id:A002580 - cp's OEIS frontend

A253904 Denominators of approximation to 2^(1/3) by Halley's method after n iterations.

1, 4, 504, 387144514512, 134785660354544802902690364367892668197456173472

Offset: 0

Kival Ngaokrajang, Jan 24 2015

Numerators are given in A253690.

			Approximations to 2^(1/3):
n = 1: 5/4 = 1.25; error = -0.00992104...
n = 2: 635/504 = 1.2599206...; error = -0.00000041...
n = 3: 487771523185/387144514512 = 1.2599210...; error = -3.001136... * 10^-20.

Eric Weisstein's World of Mathematics, Halley's method
Wikipedia, Cube root

Cf. A002580, A248041, A248042, A253690.

{a=1; b=1; print1(b, ", "); for(n=1, 5, x=a*(a^3+4*b^3); y=2*b*(a^3+b^3); a=x/gcd(x, y); b=y/gcd(x, y); print1(b, ", "))}

a(n) = y(n)/gcd(x(n),y(n))
where x(n) = A253690(n-1)*(A253690(n-1)^3n + 4*a(n-1)^3)
and y(n) = 2*(A253690(n-1)^3 + a(n-1)^3);
x(0) = y(0) = 1.

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

A100082 First occurrence of n in the fractional part of the cube root of 2, starting with n=0.

Original entry on oeis.org

7, 6, 1, 15, 8, 2, 17, 14, 10, 3, 6, 61, 62, 74, 42, 48, 16, 392, 193, 57, 346, 5, 31, 461, 90, 1, 148, 28, 32, 67, 145, 15, 328, 621, 267, 34, 92, 81, 232, 75, 126, 99, 147, 91, 268, 139, 43, 18, 12, 8, 35, 41, 66, 255, 101, 65, 88, 37, 207, 2, 25, 112, 89, 266, 17, 72, 115

Offset: 0

Mark Hudson (mrmarkhudson(AT)hotmail.com), Nov 04 2004

			The fractional part of 2^(1/3) begins 259921049894873164767210607278... so the first occurrence of 0 is at position 7; the first occurrence of 1 is at position 6; etc.

Eric Weisstein's World of Mathematics, Delian Constant

Cf. A002580 for digits of the cube root of 2.

Join[{7},With[{cr2=Rest[RealDigits[2^(1/3),10,1000][[1]]]},Flatten[ Table[Position[ Partition[cr2,IntegerLength[n],1],IntegerDigits[n],1,1],{n,70}]]]] (* Harvey P. Dale, Nov 28 2012 *)

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)

A380898 Decimal expansion of 2^(8/3).

Original entry on oeis.org

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

Offset: 1

Stefano Spezia, Feb 07 2025

This constant is the effective degree of generic random walk on the dual (4,8,8) lattice (see Burda et al.).

			6.3496042078727978990068225570892330415659733...

J. Burda, J. Duda, J. M. Luck, and B. Waclaw, The Various Facets of Random Walk Entropy, Acta Phys. Pol. B 41, 949-987 (2010). See pp. 964, 980.

Cf. A002580, A010588, A010603.

Mathematica
```
RealDigits[2^(8/3),10,100][[1]]
```

Equals A010588^2 = 2*A010603. - Hugo Pfoertner, Feb 07 2025

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A253904 Denominators of approximation to 2^(1/3) by Halley's method after n iterations.

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

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A100082 First occurrence of n in the fractional part of the cube root of 2, starting with n=0.

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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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A380898 Decimal expansion of 2^(8/3).

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