A011002 -id:A011002 - cp's OEIS frontend

A011350 Decimal expansion of 6th root of 15.

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

Offset: 1

N. J. A. Sloane

The 6th root r(6) of the expected value E(x^6) for a normal distribution with zero mean and standard deviation 1. See A289090 for more details. - Stanislav Sykora, Jul 26 2017

Ivan Panchenko, Table of n, a(n) for n = 1..1000

Cf. A011002, A076668, A289090, A289091.

RealDigits[Power[15, (6)^-1],10,120][[1]]  (* Harvey P. Dale, Apr 09 2011 *)

sqrtn(15, 6) \\ Michel Marcus, Jul 27 2017

A289090 Decimal expansion of (E(|x|^3))^(1/3), with x being a normally distributed random variable.

Original entry on oeis.org

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

Offset: 1

Stanislav Sykora, Jul 26 2017

The p-th root r(p) of the expected value E(|x|^p) for various distributions appears, for example, in chemical physics, where some interactions depend on high powers of interatomic distances.
When x is distributed normally with zero mean and standard deviation 1, r(p) evaluates to r(p) = ((p-1)!!*w(p))^(1/p), where w(p) = 1 for even p and sqrt(2/Pi) for odd p. Note that, by definition, r(2) = 1 and r(1) = w(1) = A076668.
The present constant is a = r(3).

			1.16857525496246554867047601109768527106052404816790797238351628742...

Stanislav Sykora, Table of n, a(n) for n = 1..1000
Wikipedia, Normal distribution

Cf. A060294, A076668 (p=1), A011002 (p=4), A289091 (p=5), A011350 (p=6).

ExpectedValue[Abs[#]^3&, NormalDistribution[0, 1]]^(1/3) // RealDigits[#, 10, 105]& // First (* Jean-François Alcover, Jul 28 2017 *)

\\ General code, for any p > 0:
r(p) = (sqrt(2/Pi)^(p%2)*prod(k=0,(p-2)\2,p-1-2*k))^(1/p);
a = r(3) \\ Present instance

Equals (2!!*sqrt(2/Pi))^(1/3) = (2*A076668)^(1/3).

A289091 Decimal expansion of (E(|x|^5))^(1/5), with x being a normally distributed random variable.

Original entry on oeis.org

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

Offset: 1

Stanislav Sykora, Jul 26 2017

The 5th root r(5) of the expected value E(|x|^5) for a normal distribution with zero mean and standard deviation 1. See A289090 for more details.

			1.44879190154930528525354659881281058821340103935196780729503058015...

Stanislav Sykora, Table of n, a(n) for n = 1..1000
Wikipedia, Normal distribution

Cf. A060294, A076668 (p=1), A289090 (p=3), A011002 (p=4), A011350 (p=6).

// General code, for any p > 0:
r(p) = (sqrt(2/Pi)^(p%2)*prod(k=0,(p-2)\2,p-1-2*k))^(1/p);
a = r(5) // Present instance

a = r(5), where r(p) = ((p-1)!!*sqrt(2/Pi))^(1/p).

a = (8*A076668)^(1/5).

A179615 Continued fraction for 3^(1/4).

Original entry on oeis.org

1, 3, 6, 9, 1, 1, 2, 1, 2, 1, 2, 5, 1, 12, 5, 1, 4, 1, 13, 1, 6, 1, 22, 1, 8, 21, 3, 142, 1, 1, 2, 1, 2, 2, 7, 1, 2, 1, 1, 1, 5, 3, 1, 1, 2, 1, 1, 3, 1, 1, 1, 1, 106, 1, 1, 1, 1, 1, 1, 7, 1, 22, 1, 71, 1, 4, 2, 1, 3, 1, 2, 2, 1, 4, 1, 2, 2, 2, 2, 1, 2, 82, 1, 3, 1, 4, 1, 30, 1, 3, 1, 2, 1, 1, 1, 16, 1, 2

Offset: 0

Vladimir Joseph Stephan Orlovsky, Jul 20 2010

3^(1/4) = 1.31607401295249246... (see A011002)

Cf. A011002 (decimal expansion), A179613.

Mathematica
```
ContinuedFraction[3^(1/4),200]
```

contfrac(sqrtn(3,4)) \\ Charles R Greathouse IV, Apr 14 2014

Offset changed by Andrew Howroyd, Jul 07 2024

A379414 a(n) = n + floor(ns/r) + floor(nt/r), where r = 3^(1/4), s = 3^(1/2), t = 3^(3/4).

Original entry on oeis.org

3, 7, 11, 15, 19, 23, 28, 31, 35, 40, 44, 47, 52, 56, 59, 64, 68, 72, 76, 80, 84, 88, 92, 96, 100, 105, 108, 112, 117, 120, 124, 129, 133, 136, 141, 145, 149, 153, 157, 161, 165, 169, 173, 177, 181, 185, 189, 194, 197, 201, 206, 210, 213, 218, 222, 225, 230

Offset: 1

Clark Kimberling, Jan 18 2025

nonn

This sequence and A379415 and A379416 partition the positive integers; see A184812 for a proof.
For each k in A000027, write "a" if k=A379414(n) for some n, "b" if k=A379415(n) for some n, and "c" if k=A379416(n) for some n. Concatenating these letters for k = 1,2,3,... spells the following infinite word:
cbacbcabccabcbacbcacbcabcbcacbacbcabccbacbcabcacbcbacbcacbacbcbacbcacbcabcbaccbacbcabccabcbacbcacbcabcbcacbacbcabccbacbacbcabccbacbcabcacbcba

Cf. A184812, A379415, A379416.

Cf. also A011002, A002194, A011022.

r = 3^(1/4); s = 3^(1/2); t = 3^(3/4);
Table[n + Floor[n*s/r] + Floor[n*t/r], {n, 1, 120}]  (* A379411 *)
Table[n + Floor[n*r/s] + Floor[n*t/s], {n, 1, 120}]  (* A379412 *)
Table[n + Floor[n*r/t] + Floor[n*s/t], {n, 1, 120}]  (* A379413 *)

a(n) = n + floor(n*r) + floor(n*r^2), where r = 3^(1/4).

A379415 a(n) = n + floor(nr/s) + floor(nt/s), where r = 3^(1/4), s = 3^(1/2), t = 3^(3/4).

Original entry on oeis.org

2, 5, 8, 12, 14, 17, 21, 24, 26, 30, 33, 36, 39, 42, 45, 49, 51, 54, 58, 61, 63, 66, 70, 73, 75, 79, 82, 85, 89, 91, 94, 98, 101, 103, 107, 110, 113, 116, 119, 122, 125, 128, 131, 134, 138, 140, 143, 147, 150, 152, 156, 159, 162, 166, 168, 171, 175, 178, 180

Offset: 1

Clark Kimberling, Jan 18 2025

nonn

This sequence and A379414 and A379416 partition the positive integers; see A184812 for a proof.

Cf. A184812, A379415, A379416.

Cf. also A011002, A002194, A011022.

r = 3^(1/4); s = 3^(1/2); t = 3^(3/4);
Table[n + Floor[n*s/r] + Floor[n*t/r], {n, 1, 120}]  (* A379414 *)
Table[n + Floor[n*r/s] + Floor[n*t/s], {n, 1, 120}]  (* A379415 *)
Table[n + Floor[n*r/t] + Floor[n*s/t], {n, 1, 120}]  (* A379416 *)

a(n) = n + floor(n/r) + floor(n*r), where r = 3^(1/4).

A093604 Decimal expansion of D/2, where D^2 = 3*sqrt(3)/Pi.

Original entry on oeis.org

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

Offset: 0

Lekraj Beedassy, May 14 2004

D/2=sqrt(3*sqrt(3)/Pi)/2 corresponds to the radius of the area-bisecting concentric circle within the unit-sided hexagon.

			sqrt(3*sqrt(3)/Pi)/2 = 0.6430370685787437846417825056651579788623049833326304871239...

Index entries for transcendental numbers

Cf. A097603, A010527, A011002, A087197. - R. J. Mathar, Feb 06 2009

RealDigits[Sqrt[(3Sqrt[3])/Pi]/2,10,120][[1]] (* Harvey P. Dale, Aug 27 2017 *)

sqrt(sqrt(27)/Pi)/2 \\ Charles R Greathouse IV, Oct 01 2022

Removed leading zero and adjusted offset - R. J. Mathar, Feb 06 2009

Corrected and extended by Harvey P. Dale, Aug 27 2017

A154605 Decimal expansion of 2/(4th root of 3).

Original entry on oeis.org

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

Offset: 1

Rick L. Shepherd, Jan 12 2009

The side length of an equilateral triangle with area 1.

Quartic number with denominator 3 and minimal polynomial 3x^4 - 16. - Charles R Greathouse IV, Jun 30 2021

			1.5196713713031850946623755013090906...

Cf. A011002, A120011.

RealDigits[N[2/3^(1/4),200]][[1]] (* Vladimir Joseph Stephan Orlovsky, Feb 05 2012 *)
RealDigits[2/Surd[3,4],10,120][[1]] (* Harvey P. Dale, Mar 09 2019 *)

PARI
```
2/3^(1/4)
```
PARI
```
2/sqrt(sqrt(3))
```

A154605 = 2/A011002 = 2/(3^(1/4)).

A268604 Decimal expansion of 6/sqrt(sqrt(3)).

Original entry on oeis.org

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

Offset: 1

Stanislav Sykora, Feb 08 2016

This is the smallest possible perimeter index P/sqrt(A), with P being the perimeter and A the enclosed area, among all triangles, and sets of triangles, in the Euclidean plane. The minimum value is attained by a single equilateral triangle. See also A019707.

			4.559014113909555283987126503927272012380640693323861911667731922...

Stanislav Sykora, Table of n, a(n) for n = 1..2000

Cf. A011002, A019707.

RealDigits[6/Sqrt[Sqrt[3]], 10, 100][[1]] (* Bruno Berselli, Feb 09 2016 *)

PARI
```
a=6/sqrt(sqrt(3))
```

A011042 Decimal expansion of 4th root of 48.

Original entry on oeis.org

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

Offset: 1

N. J. A. Sloane

Ivan Panchenko, Table of n, a(n) for n = 1..1000

RealDigits[Power[48, (4)^-1],10,120][[1]] (* Harvey P. Dale, Jul 26 2011 *)

sqrtn(48,4) \\ Charles R Greathouse IV, Apr 15 2014

Equals 2*A011002. [From R. J. Mathar, Feb 04 2009]

cp's OEIS Frontend

A011350 Decimal expansion of 6th root of 15.

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A289090 Decimal expansion of (E(|x|^3))^(1/3), with x being a normally distributed random variable.

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A289091 Decimal expansion of (E(|x|^5))^(1/5), with x being a normally distributed random variable.

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A179615 Continued fraction for 3^(1/4).

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A379414 a(n) = n + floor(n*s/r) + floor(n*t/r), where r = 3^(1/4), s = 3^(1/2), t = 3^(3/4).

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A379415 a(n) = n + floor(n*r/s) + floor(n*t/s), where r = 3^(1/4), s = 3^(1/2), t = 3^(3/4).

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A093604 Decimal expansion of D/2, where D^2 = 3*sqrt(3)/Pi.

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A154605 Decimal expansion of 2/(4th root of 3).

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A379414 a(n) = n + floor(ns/r) + floor(nt/r), where r = 3^(1/4), s = 3^(1/2), t = 3^(3/4).

A379415 a(n) = n + floor(nr/s) + floor(nt/s), where r = 3^(1/4), s = 3^(1/2), t = 3^(3/4).