A074457 -id:A074457 - cp's OEIS frontend

A074455 Consider the volume of unit sphere as a function of the dimension d; maximize this as a function of d (considered as a continuous variable); sequence gives decimal expansion of the best d.

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

Offset: 1

Robert G. Wilson v, Aug 22 2002

From David W. Wilson, Jul 12 2007: (Start)
For an integer d, the volume of a d-dimensional unit ball is v(d) = Pi^(d/2)/(d/2)! and its surface area is area(d) = d*Pi^(d/2)/(d/2)! = d*v(d). If we interpolate n! = gamma(n+1) we can define v(d) and area(d) as continuous functions for (at least) d >= 0.
A074457 purports to minimize area(d). Since area(d+2) = 2*Pi*v(d), area() is minimized at y = x+2; therefore A074457 coincides with the current sequence except at the first term. (End)

			5.256946404860576780132838388690769236619017237183214857509879678777109...

J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer-Verlag, p. 9.
Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Section 1.5.4, p. 34.
David Wells, The Penguin Dictionary of Curious and Interesting Numbers. Penguin Books, NY, 1986, Revised edition 1987. See p. 67.

Brian Hayes, An Adventure in the Nth Dimension, pp. 30-42 of M. Pitici, editor, The Best Writing on Mathematics 2012, Princeton Univ. Press, 2012. See p. 42. - From _N. J. A. Sloane_, Jan 13 2013; alternative link.
Eric Weisstein's World of Mathematics, Ball.

Cf. A074457.

The volume is given by A074454. Cf. A072345 & A072346.

x /. FindRoot[ PolyGamma[1 + x/2] == Log[Pi], {x, 5}, WorkingPrecision -> 105] // RealDigits // First (* Jean-François Alcover, Mar 28 2013 *)

hyperspheresurface(d)=2*Pi^(d/2)/gamma(d/2)
hyperspherevolume(d)=hyperspheresurface(d)/d
FindMax(fn_x,lo,hi)=
{
local(oldprecision, x, y, z);
oldprecision = default(realprecision);
default(realprecision, oldprecision+10);
while (hi-lo > 10^-oldprecision,
while (1,
z = vector(2, i, lo*(3-i)/3 + hi*i/3);
y = vector(2, i, eval(Str("x = z[" i "]; " fn_x)));
if (abs(y[1]-y[2]) > 10^(5-default(realprecision)), break);
default(realprecision, default(realprecision)+10);
);
if (y[1] < y[2], lo = z[1], hi = z[2]);
);
default(realprecision, oldprecision);
(lo + hi) / 2.
}
default(realprecision, 105);
A074455=FindMax("hyperspherevolume(x)", 1, 9)
A074457=FindMax("hyperspheresurface(x)", 1, 9)
A074454=hyperspherevolume(A074455)
A074456=hyperspheresurface(A074457)
/* David W. Cantrell */

2 * (solve(x=3, 4, psi(x) - log(Pi)) - 1) \\ Jianing Song, May 12 2025

d = root of Psi((1/2)*d + 1) = log(Pi).

d is 2 less than the number with decimal digits A074457 (the hypersphere dimension that maximizes hypersurface area). - Eric W. Weisstein, Dec 02 2014

Corrected by Eric W. Weisstein, Aug 31 2003

Corrected by Martin Fuller, Jul 12 2007

A164081 Floor of 2^(n-1) times the surface area of the unit sphere in 2n-dimensional space.

Original entry on oeis.org

6, 39, 124, 259, 408, 512, 536, 481, 378, 264, 166, 94, 49, 24, 10, 4, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0

Offset: 1

Jonathan Vos Post, Aug 09 2009

nonn

The rounded values of this real sequence is A164082, the ceiling is A164083.
The surface area of n-dimensional sphere of radius r is n*V_n*r^(n-1); see A072478/A072479.
There are only 17 nonzero terms. - G. C. Greubel, Sep 10 2017

			Table of approximate real values before taking integer part.
========================
n (2*Pi)^n / (n-1)!
1 6.28318531 = A019692
2 39.4784176 = 2*A164102
3 124.025107 = 4*A091925
4 259.757576 = 8*A164109
5 408.026246
6 512.740903
7 536.941018
8 481.957131
9 378.528246
10 264.262568
11 166.041068
12 94.8424365
13 49.6593836
14 24.00147
15 10.7718345
16 4.5120955
17 1.77189576
18 0.654891141
19 0.228600133
20 0.075596684
========================

J. H. Conway and N. J. A. Sloane, Sphere Packings, Lattices, and Groups, 2nd ed., New York: Springer-Verlag, p. 9, 1993.
H. S. M. Coxeter, Regular Polytopes, 3rd ed., New York: Dover, 1973.
D. M. Y. Sommerville, An Introduction to the Geometry of n Dimensions, New York: Dover, p. 136, 1958.

Eric W. Weisstein, Hypersphere.

Cf. A072345, A072346, A072478, A072479, A074457, A122510, A154255.

A164081 := proc(n) (2*Pi)^n/(n-1)! ; floor(%) ; end: seq(A164081(n),n=1..80) ; # R. J. Mathar, Sep 09 2009

Table[Floor[(2*Pi)^n/(n - 1)!], {n, 1, 100}] (* G. C. Greubel, Sep 10 2017 *)

for(n=1,100, print1(floor((2*Pi)^n/(n-1)!), ", ")) \\ G. C. Greubel, Sep 10 2017

a(n) = floor( (2*Pi)^n/(n-1)! ).

Definition corrected by R. J. Mathar, Sep 09 2009

A164082 Rounded value of 2^(n-1) times the surface area of the unit sphere in 2n-dimensional space.

Original entry on oeis.org

6, 39, 124, 260, 408, 513, 537, 482, 379, 264, 166, 95, 50, 24, 11, 5, 2, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0

Offset: 1

Jonathan Vos Post, Aug 09 2009

nonn

The floor of this real sequence is A164081, the ceiling is A164083.
The surface area of the n-dimensional sphere of radius r is n*V_n*r^(n-1); see A072478/ A072479.
There are 18 nonzero terms in this sequence. - G. C. Greubel, Sep 11 2017

			Table of approximate real values before rounding up or down:
========================
n ((2*pi)^n) / (n-1)!
1 6.28318531 = A019692
2 39.4784176 = 2*A164102
3 124.025107 = 4*A091925
4 259.757576 = 8*A164109
5 408.026246
6 512.740903
7 536.941018
8 481.957131
9 378.528246
10 264.262568
11 166.041068
12 94.8424365
13 49.6593836
14 24.00147
15 10.7718345
16 4.5120955
17 1.77189576
18 0.654891141
19 0.228600133
20 0.075596684
========================

Conway, J. H. and Sloane, N. J. A. Sphere Packings, Lattices, and Groups, 2nd ed. New York: Springer-Verlag, p. 9, 1993.
Coxeter, H. S. M. Regular Polytopes, 3rd ed. New York: Dover, 1973.
Sommerville, D. M. Y. An Introduction to the Geometry of n Dimensions. New York: Dover, p. 136, 1958.

Eric W. Weisstein, Hypersphere,

Cf. A072345, A072346, A072478, A072479, A074457, A122510, A154255, A164081, A164083.

A164082 := proc(n) (2*Pi)^n/(n-1)! ; round(%) ; end: seq(A164082(n),n=1..80) ; # R. J. Mathar, Sep 09 2009

Table[Round[(2*Pi)^n/(n - 1)!], {n, 1, 20}] (* G. C. Greubel, Sep 11 2017 *)

for(n=1,20, print1(round((2*Pi)^n/(n-1)!), ", ")) \\ G. C. Greubel, Sep 11 2017

a(n) = round(((2*Pi)^n)/(n-1)!).

Definition corrected by R. J. Mathar, Sep 09 2009

A164083 Ceiling of 2^(n-1) times the surface area of the unit sphere in 2n-dimensional space.

Original entry on oeis.org

7, 40, 125, 260, 409, 513, 537, 482, 379, 265, 167, 95, 50, 25, 11, 5, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1

Offset: 1

Jonathan Vos Post, Aug 09 2009

nonn

The rounded values of this real sequence is A164082, the floor is A164081.

The surface area of n-dimensional sphere of radius r is n*V_n*r^(n-1); see A072478/A072479.

			Table of approximate real values before rounding up.
========================
n ((2*pi)^n) / (n-1)!
1 6.28318531 = A019692
2 39.4784176 = 2*A164102
3 124.025107 = 4*A091925
4 259.757576 = 8*A164109
5 408.026246
6 512.740903
7 536.941018
8 481.957131
9 378.528246
10 264.262568
11 166.041068
12 94.8424365
13 49.6593836
14 24.00147
15 10.7718345
16 4.5120955
17 1.77189576
18 0.654891141
19 0.228600133
20 0.075596684
========================

Conway, J. H. and Sloane, N. J. A. Sphere Packings, Lattices, and Groups, 2nd ed. New York: Springer-Verlag, p. 9, 1993.
Coxeter, H. S. M. Regular Polytopes, 3rd ed. New York: Dover, 1973.
Sommerville, D. M. Y. An Introduction to the Geometry of n Dimensions. New York: Dover, p. 136, 1958.

Eric W. Weisstein, Hypersphere,

Cf. A072345, A072346, A072478, A072479, A074457, A122510, A154255, A164081, A164082.

Table[Ceiling[(2Pi)^n/(n-1)!],{n,60}] (* Harvey P. Dale, Jul 30 2020 *)

a(n) = ceiling(((2*pi)^n)/(n-1)!).

Definition corrected - R. J. Mathar, Sep 09 2009

A175477 Decimal expansion of the dimension in which the sphere of unit radius has unit volume.

Original entry on oeis.org

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

Offset: 2

R. J. Mathar, May 25 2010

The positive solution x to Pi^(x/2)/Gamma(x/2+1) = 1.
Then Pi^(x/2) = 1488.75641500529701...
From Mohammed Yaseen, Sep 25 2022: (Start)
0 is another solution. All other solutions are negative.
This is also the dimension d in which the sphere of unit radius has surface area d. (End)

			12.764052935032681271263280950768574761998...

Mohammed Yaseen, Table of n, a(n) for n = 2..10000
Wikipedia, N-sphere

Cf. A074455, A074457.

x /. FindRoot[ Pi^(x/2)/Gamma[x/2 + 1] == 1, {x, 12}, WorkingPrecision -> 105] // RealDigits[#, 10, 105] & // First (* Jean-François Alcover, Feb 12 2013 *)

solve(x=9,13,Pi^(x/2)-gamma(x/2+1)) \\ Charles R Greathouse IV, Jan 30 2016

A244619 Decimal expansion of 'theta', the unique positive root of the equation polygamma(x) = log(Pi), where polygamma(x) gives gamma'(x)/gamma(x), that is the logarithmic derivative of the gamma function.

Original entry on oeis.org

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

Offset: 1

Jean-François Alcover, Jul 02 2014

This constant appears in d_a = 2*theta = 7.2569464... and d_v = 2*(theta-1) = 5.2569464..., the fractional dimensions at which d-dimensional spherical surface area and volume, respectively, are maximized. [after Steven Finch]

			3.6284732024302883900664191943453846183...

Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 1.5.4 Gamma Function, p. 34.

Cf. A074454, A074455, A074456, A074457.

theta = x /. FindRoot[PolyGamma[x] == Log[Pi], {x, 4}, WorkingPrecision -> 105]; RealDigits[theta] // First

polygamma(n, x) = if (n == 0, psi(x), (-1)^(n+1)*n!*zetahurwitz(n+1, x));
solve(x=3.5, 3.7, polygamma(0, x) - log(Pi)) \\ Gheorghe Coserea, Sep 30 2018

cp's OEIS Frontend

A074455 Consider the volume of unit sphere as a function of the dimension d; maximize this as a function of d (considered as a continuous variable); sequence gives decimal expansion of the best d.

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A164081 Floor of 2^(n-1) times the surface area of the unit sphere in 2n-dimensional space.

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A164082 Rounded value of 2^(n-1) times the surface area of the unit sphere in 2n-dimensional space.

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A164083 Ceiling of 2^(n-1) times the surface area of the unit sphere in 2n-dimensional space.

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A175477 Decimal expansion of the dimension in which the sphere of unit radius has unit volume.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

A244619 Decimal expansion of 'theta', the unique positive root of the equation polygamma(x) = log(Pi), where polygamma(x) gives gamma'(x)/gamma(x), that is the logarithmic derivative of the gamma function.

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Author

Keywords

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Examples

References

Crossrefs

Programs

Mathematica