A002580 -id:A002580 - cp's OEIS frontend

A010769 Decimal expansion of 7th root of 2.

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

Offset: 1

N. J. A. Sloane

This is also the unique positive attractor of the mapping M(x) = sqrt(sqrt(sqrt(2*x))). In general, (p^N-1)-th root of a number f can be approximated by iterating the mapping M(x) = (f*x)^(1/p^N). The convergence is very fast. In this case, p=2, N=3, and f=2. In the form "evaluate the 3rd (or 7th or 15th) root of a number using only square roots", the insight is usable as a recreational math puzzle. - Stanislav Sykora, Oct 26 2015

			1.104089513673812337649505387623...

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

Cube roots (p=2,N=2) for various f: A002580 (2), A002581 (3), A005480 (4), A010582 (10), A092041 (e). 7th roots (p=2,N=3): A246709 (3), A011186 (4), A011201 (5), A011276 (10), A092516 (e). 8th roots (p=3,N=2): A010770 (2), A246710 (3), A011202 (5), A011277 (10). 15th roots (p=2,N=4): A010777(2), A011194(4), A011209(5), A011284(10). - Stanislav Sykora, Oct 26 2015

RealDigits[N[2^(1/7), 100]][[1]] (* Vincenzo Librandi, Apr 02 2013 *)
RealDigits[Surd[2,7],10,120][[1]] (* Harvey P. Dale, Sep 05 2022 *)

sqrtn(2,7) \\ Charles R Greathouse IV, Apr 15 2014

{ default(realprecision, 100); x= 2^(1/7); for(n=1, 100, d=floor(x); x=(x-d)*10; print1(d, ", ")) } \\ Altug Alkan, Nov 14 2015

Equals Product_{k>=0} (1 + (-1)^k/(7*k + 6)). - Amiram Eldar, Jul 29 2020

A010581 Decimal expansion of cube root of 9.

Original entry on oeis.org

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

Offset: 1

N. J. A. Sloane

			2.08008382305190411453005682435788538633780534037326210...

Horace S. Uhler, Many-figure approximations for cubic root of 2, cubic root of 3, cubic root of 4 and cubic root of 9 with chi 2 data, Scripta Math. 18, (1952). 173-176.

Harry J. Smith, Table of n, a(n) for n = 1..20000
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]

Cf. A010239 = Continued fraction. - Harry J. Smith, May 07 2009

RealDigits[N[9^(1/3), 200]][[1]] (* Vladimir Joseph Stephan Orlovsky, Jan 24 2012 *)

{ default(realprecision, 20080); x=9^(1/3); for (n=1, 20000, d=floor(x); x=(x-d)*10; write("b010581.txt", n, " ", d)); } \\ Harry J. Smith, May 07 2009, corrected May 19 2009

Revised by N. J. A. Sloane, Apr 23 2006

A235362 Decimal expansion of the cube root of 2 divided by 2.

Original entry on oeis.org

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

Offset: 0

Alonso del Arte, Jan 07 2014

Also reciprocal of the real cubic root of 4 and negated real part of either complex cubic root of 2.

			0.6299605249474365823836053...

Richard Zippel, The Weyl Computer Algebra Substrate in Design and Implementation of Symbolic Computation Systems: International Symposium, DISCO '93 Gmunden, Austria, September 15-17, 1993. Proceedings, p. 306.

Cf. A002580, A010503, A020761, A239797.

Digits := 100 ; evalf(1/2^(2/3)) ; # R. J. Mathar, Jan 16 2023

Mathematica
```
RealDigits[1/2^(2/3), 10, 128][[1]]
```

sqrtn(1/4,3) \\ 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 1/A005480 = A002580 /2 . - Wolfdieter Lang, Jan 02 2023

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

Original entry on oeis.org

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

A290570 Decimal expansion of Integral_{0..Pi/2} dtheta/(cos(theta)^3 + sin(theta)^3)^(2/3).

Original entry on oeis.org

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

Offset: 1

Jean-François Alcover, Aug 07 2017

			1.766638750285449957313689499648438702571868538202557530126905241835453...

Oscar S. Adams, Elliptic Functions Applied to Conformal World Maps, Special Publication No. 112 of the U.S. Coast and Geodetic Survey, 1925. See constant K p. 9 and previous pages.

StackExchange, Question 2380131 Evaluate in closed form Integral_{0..Pi/2} dtheta/(cos(theta)^3+sin(theta)^3)^(2/3), August 2017.

Cf. A073005 (Gamma(1/3)), A073006 (Gamma(2/3)), A197374 (Beta(1/3,1/3)).

Cf. A104133, A104134, A197374.

RealDigits[(1/3)*Gamma[1/3]^2/Gamma[2/3], 10, 105]

(1/3)*gamma(1/3)^2/gamma(2/3) \\ Michel Marcus, Aug 07 2017

Equals (1/3)*Beta(1/3,1/3).
Equals (1/3)*Gamma(1/3)^2/Gamma(2/3).
Equals A197374/3. - Michel Marcus, Jun 08 2020
From Peter Bala, Mar 01 2022: (Start)
Equals 2*Sum_{n >= 0} (1/(3*n+1) + 1/(3*n-2))*binomial(1/3,n). Cf. A002580 and A175576.
Equals Sum_{n >= 0} (-1)^n*(1/(3*n+1) - 1/(3*n-2))*binomial(1/3,n).
Equals hypergeom([1/3, 2/3], [4/3], 1) = (3/2)*hypergeom([-1/3, -2/3], [4/3], 1) = 2*hypergeom([1/3, 2/3], [4/3], -1) = hypergeom([-1/3, -2/3, 5/6], [4/3, -1/6], -1). (End)

A098328 Recurrence sequence derived from the digits of the cube root of 2 after its decimal point.

Original entry on oeis.org

0, 7, 14, 42, 147, 321, 473, 322, 785, 1779, 3039, 1957, 16446, 274134, 374781, 110639, 248175, 385504, 2359264, 5108010, 3822244, 3812946, 9896631

Offset: 0

Mark Hudson (mrmarkhudson(AT)hotmail.com), Sep 14 2004

			2^(1/3)=1.259921049894873164767210607...
So for example, with a(1)=0, a(2)=7 because the 7th digit after the decimal point is 0; a(3)=14 because the 14th digit after the decimal point is 7 and so on.

Other recurrence sequences: A097614 for Pi, A098266 for e, A098289 for log(2), A098290 for Zeta(3), A098319 for 1/Pi, A098320 for 1/e, A098321 for gamma, A098322 for G, A098323 for 1/G, A098324 for Golden Ratio (phi), A098325 for sqrt(Pi), A098326 for sqrt(2), A098327 for sqrt(e). A002580 for digits of 2^(1/3).

with(StringTools): Digits:=10000: G:=convert(evalf(root(2,3)),string): a[0]:=0: for n from 1 to 12 do a[n]:=Search(convert(a[n-1],string), G)-2:printf("%d, ",a[n-1]):od: # Nathaniel Johnston, Apr 30 2011

a(1)=0. a(1)=0, p(i)=position of first occurrence of a(i) in decimal places of 2^(1/3), a(i+1)=p(i).

More terms from Ryan Propper, Jul 21 2006

A248041 Numerators of approximation to 2^(1/3) by Newton's method after n iterations.

Original entry on oeis.org

1, 4, 91, 1126819, 2146097524939083451

Offset: 0

Kival Ngaokrajang, Jan 11 2015

Denominators are given in A248042.

			Approximations to 2^(1/3):
n = 1:            4/3 = 1.33333...; error = 0.07341...
n = 2:          91/72 = 1.26388...; error = 0.00396...
n = 3: 1126819/894348 = 1.25993...; error = 0.00001...
...

Eric Weisstein's World of Mathematics, Cube Root
Wikipedia, Cube Root

Cf. A002580, A248042.

a(n) = x(n)/gcd(x(n),y(n))
where x(n) = 2*(a(n-1)^3*A248042(n-1)^2 + A248042(n-1)^5)
and y(n) = 3*a(n-1)^2*A248042(n-1)^3;
x(0) = y(0) = 1.

A248042 Denominators of approximation to 2^(1/3) by Newton's method after n iterations.

Original entry on oeis.org

1, 3, 72, 894348, 1703358734191174242

Offset: 0

Kival Ngaokrajang, Jan 11 2015

Numerators are given in A248041.

			Approximations to 2^(1/3):
n = 1:            4/3 = 1.33333...; error = 0.07341...
n = 2:          91/72 = 1.26388...; error = 0.00396...
n = 3: 1126819/894348 = 1.25993...; error = 0.00001...
...

Eric Weisstein's World of Mathematics, Cube Root
Wikipedia, Cube Root

Cf. A002580, A248041.

a(n) = y(n)/gcd(x(n),y(n))
where x(n) = 2*(A248041(n-1)^3*a(n-1)^2 + a(n-1)^5)
and y(n) = 3*A248041(n-1)^2*a(n-1)^3;
x(0) = y(0) = 1.

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

A010588 Decimal expansion of cube root of 16.

Original entry on oeis.org

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

Offset: 1

N. J. A. Sloane

			2.51984209978974632953442121455645670114050292940301...

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

Cf. A002580.

RealDigits[N[16^(1/3),200]][[1]] (* Vladimir Joseph Stephan Orlovsky, Feb 06 2012 *)
RealDigits[CubeRoot[16],10,120][[1]] (* Harvey P. Dale, Apr 24 2023 *)

sqrtn(16,3) \\ Charles R Greathouse IV, Apr 14 2014

Equals 2 * A002580. - Amiram Eldar, Jun 25 2023

cp's OEIS Frontend

A010769 Decimal expansion of 7th root of 2.

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A010581 Decimal expansion of cube root of 9.

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A235362 Decimal expansion of the cube root of 2 divided by 2.

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

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A290570 Decimal expansion of Integral_{0..Pi/2} dtheta/(cos(theta)^3 + sin(theta)^3)^(2/3).

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A098328 Recurrence sequence derived from the digits of the cube root of 2 after its decimal point.

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Maple

Formula

Extensions

A248041 Numerators of approximation to 2^(1/3) by Newton's method after n iterations.

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