A002580 -id:A002580 - cp's OEIS frontend

A138374 Count of post-period decimal digits up to which the rounded n-th convergent to 2^(1/3) agrees with the exact value.

1, 1, 2, 2, 3, 4, 4, 6, 6, 8, 6, 10, 10, 12, 13, 15, 16, 17, 16, 18, 19, 20, 21, 21, 22, 23, 25, 27, 28, 29, 30, 31, 32, 35, 38, 39, 40, 39, 41, 42, 45, 46, 46, 47, 49, 51, 52, 52, 54, 56, 56, 57, 58, 58, 60, 61, 62, 63, 65, 64, 66, 68, 69, 69, 70, 70, 72, 74, 74, 75, 77, 79, 81

Offset: 1

Artur Jasinski, Mar 17 2008

This is a measure of the quality of the n-th convergent to the constant A002580 if the convergent and the exact value are compared rounded to an increasing number of digits. The sequence of rounded values of A002580 is 1, 1.3, 1.26, 1.260, 1.2599, 1.25992, 1.259921, 1.2599211 etc. The n-th convergents are taken from A002352 and A002351, each with associated rounded decimal expansions.

a(n) is the maximum number of post-period digits of the two expansions if compared at the same level of rounding.

			For n=5, the 5th convergent is 63/50 = 1.26000000.., with a sequence of rounded representations 1, 1.3, 1.26, 1.260, 1.2600, 1.26000, etc.
Rounded to 1, 2, or 3 post-period decimal digits, this is the same as the rounded version of the exact value, but disagrees if both are rounded to 4 decimal digits, where 1.2599 <> 1.2600.
So a(5) = 3 (digits), the maximum rounding level with agreement.

Cf. A138335 - A138339, A138343, A138366, A138367, A138369 - A138373.

Definition and values replaced as defined via continued fractions - R. J. Mathar, Oct 01 2009

A246644 Decimal expansion of the real root of s^3 - s^2 + s - 1/3 = 0.

Original entry on oeis.org

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

Offset: 1

Wolfdieter Lang, Sep 02 2014

In the origami solution of doubling the cube (see the W. Lang link, p. 14, and a Sep 02 2014 comment on A002580) (1-s)/s = 2^{1/3}, or s^3 - s^2 + s - 1/3 = 0 appears, which has the real solution s = (2^(2/3) - 2^(1/3) +1)/3. In the link s is the length of the line segment(B,C') shown in Figure 16 on p. 14.

A cubic number with denominator 3. - Charles R Greathouse IV, Aug 26 2017

Cf. A002580.

First[RealDigits[(2^(2/3) - 2^(1/3) + 1)/3, 10, 100]] (* Paolo Xausa, Jun 25 2024 *)

polrootsreal(3*x^3-3*x^2+3*x-1)[1] \\ Charles R Greathouse IV, Aug 26 2017

s = 0.442493334024442103328... See the comment above.

A253583 Decimal expansion of cube root of 2 multiplied by square root of 3.

Original entry on oeis.org

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

Offset: 1

Alonso del Arte, Jan 04 2015

Multiplied by i or -i, imaginary part of either complex cube root of 16.

2^(1/3) sqrt(3) = distance between the critical points of xy(x+y)=1. - Clark Kimberling, Oct 05 2020

			2.18224727194344280712014522837961776265174667748060188140728214647356...

Chai Wah Wu, Table of n, a(n) for n = 1..10000

Cf. A002194, A002580.

RealDigits[2^(1/3) Sqrt[3], 10, 100][[1]]

sqrtn(2, 3)*sqrt(3) \\ Michel Marcus, Oct 18 2016

(-2^(1/3) + 2^(1/3)sqrt(-3))^3 = 16.

Equals A002580 * A002194. - Omar E. Pol, Jan 04 2015

A319034 Decimal expansion of the height that minimizes the total surface area of the four triangular faces of a square pyramid of unit volume.

Original entry on oeis.org

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

Offset: 1

Jon E. Schoenfield, Oct 22 2018

A square pyramid with a height of h and a base of size s X s has volume V = (1/3)*s^2*h, so a square pyramid of unit volume has s = sqrt(3/h), and the slant height of each of the four triangular faces is t = sqrt(h^2 + (s/2)^2) = sqrt(h^2 + 3/(4*h)), and the total area of the four faces is A = 4*(s*t/2) = sqrt(12*h^3 + 9)/h; this area is minimized at h = (3/2)^(1/3), where it reaches A = 3^(7/6)*2^(1/3).

If the total surface area of all five faces including the square base is to be minimized, then the resulting height is 6^(1/3) (cf. A005486). - Jon E. Schoenfield, Nov 11 2018

			1.14471424255333186780804221193967700891590692078793...

Eric Weisstein's World of Mathematics, Pyramid.

Cf. A002580, A002581, A005486, A010584.

RealDigits[Surd[3/2, 3], 10, 120][[1]] (* Amiram Eldar, Jun 21 2023 *)

sqrtn(3/2, 3) \\ Michel Marcus, Oct 23 2018

Equals (3/2)^(1/3) = (1/2)*A010584.

Equals A002581/A002580. - Michel Marcus, Oct 23 2018

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

A365932 a(n) = the number of cubes (of integers > 0) that have bit length n.

Original entry on oeis.org

1, 0, 0, 1, 1, 0, 2, 1, 1, 3, 2, 3, 5, 5, 6, 9, 10, 13, 17, 21, 26, 34, 42, 52, 67, 84, 105, 134, 167, 211, 267, 335, 422, 533, 670, 845, 1065, 1341, 1690, 2130, 2682, 3380, 4259, 5365, 6760, 8518, 10730, 13520, 17035, 21461, 27040, 34069, 42923, 54080, 68137, 85847

Offset: 1

Karl-Heinz Hofmann, Oct 05 2023

Number of cubes in the range: 2^(n-1) <= k^3 < 2^n-1.

There is no need to include 2^n-1 because it is a Mersenne number and it cannot be a power anyway.

			For n = 13; a(n) = 5; following 5 cubes have a bit length of 13: 16^3, 17^3, 18^3, 19^3 and 20^3.

Cf. A004632.
Cf. A017981 (partial sums).
Cf. A002580, A126726, A365930.

a[n_] := Floor[Surd[2^n-1, 3]] - Floor[Surd[2^(n-1)-1, 3]]; Array[a, 56] (* Amiram Eldar, Oct 30 2023 *)

from sympy import integer_nthroot
def A365932(n):
    return integer_nthroot(2**n-1, 3)[0] - integer_nthroot(2**(n-1)-1, 3)[0]
print([A365932(n) for n in range(1,57)])

a(n) = floor((2^n-1)^(1/3)) - floor((2^(n-1)-1)^(1/3)) for n > 0.

Limit_{n->oo} a(n)/a(n-1) = 2^(1/3) = A002580.

A059178 Engel expansion of 2^(1/3) = 1.25992.

Original entry on oeis.org

1, 4, 26, 32, 58, 1361, 4767, 22303, 134563, 188609, 282816, 979804, 1272032, 1330628, 3719474, 5039143, 12531368, 435451235, 5391276884, 6140156718, 24140682996, 30267765913, 56443830660, 176797839116, 645251112512

Offset: 1

Mitch Harris

Cf. A006784 for definition of Engel expansion.

F. Engel, Entwicklung der Zahlen nach Stammbruechen, Verhandlungen der 52. Versammlung deutscher Philologen und Schulmaenner in Marburg, 1913, pp. 190-191.

G. C. Greubel, Table of n, a(n) for n = 1..1000
F. Engel, Entwicklung der Zahlen nach Stammbruechen, Verhandlungen der 52. Versammlung deutscher Philologen und Schulmaenner in Marburg, 1913, pp. 190-191. English translation by Georg Fischer, included with his permission.
P. Erdős and Jeffrey Shallit, New bounds on the length of finite Pierce and Engel series, Sem. Theor. Nombres Bordeaux (2) 3 (1991), no. 1, 43-53.
Index entries for sequences related to Engel expansions

Cf. A002580.

EngelExp[A_, n_] := Join[Array[1 &, Floor[A]], First@Transpose@ NestList[{Ceiling[1/Expand[#[[1]] #[[2]] - 1]], Expand[#[[1]] #[[2]] - 1]/1} &, {Ceiling[1/(A - Floor[A])], (A - Floor[A])/1}, n - 1]];
EngelExp[N[2^(1/3), 7!], 10] (* modified by G. C. Greubel, Dec 26 2016 *)

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).

A253690 Numerators of approximation to 2^(1/3) by Halley's method after n iterations.

Original entry on oeis.org

1, 5, 635, 487771523185, 169819290704671870437365746682881808313592465345

Offset: 0

Kival Ngaokrajang, Jan 24 2015

Denominators are given in A253904.

			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, A253904.

{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(a, ", "))}

a(n) = x(n)/gcd(x(n),y(n))
where x(n) = a(n-1)*(a(n-1)^3n + 4*A253904(n-1)^3)
and y(n) = 2*(a(n-1)^3 + A253904(n-1)^3);
x(0) = y(0) = 1.

cp's OEIS Frontend

A138374 Count of post-period decimal digits up to which the rounded n-th convergent to 2^(1/3) agrees with the exact value.

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A246644 Decimal expansion of the real root of s^3 - s^2 + s - 1/3 = 0.

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A253583 Decimal expansion of cube root of 2 multiplied by square root of 3.

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A319034 Decimal expansion of the height that minimizes the total surface area of the four triangular faces of a square pyramid of unit volume.

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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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A365932 a(n) = the number of cubes (of integers > 0) that have bit length n.

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A059178 Engel expansion of 2^(1/3) = 1.25992.

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