A074455 -id:A074455 - cp's OEIS frontend

A087300 Duplicate of A074455.

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

Offset: 1

dead

A074457 Consider surface area 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.

7, 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

			7.256946404860576780132838388690769236619017237183214857509879678777...

Nenad Cakic, Dusko Letic, and Branko Davidovic, The Hyperspherical functions of a derivative, Abstr. Appl. Anal. (2010) 364292 doi:10.1155/2010/364292
Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Section 1.5.4, p. 34.

Dusko Letic, Nenad Cakic, Branko Davidovic, and Ivana Berkovic, Orthogonal and diagonal dimension fluxes of hyperspherical function, Advances in Difference Equations 2012, 2012:22. - From _N. J. A. Sloane_, Sep 04 2012
Eric Weisstein's World of Mathematics, Hypersphere.

Surface area is A074456. Cf. A072478, A072479, A074455.

RealDigits[ FindMinimum[ -n*Pi^(n/2)/(n/2)!, {n, 7}, WorkingPrecision -> 125] [[2, 1, 2]]] [[1]]
x /. FindRoot[ PolyGamma[x/2] == Log[Pi], {x, 7}, WorkingPrecision -> 105] // RealDigits // First (* Jean-François Alcover, Mar 28 2013 *)

Equals 2 + A074455.

Corrected by Eric W. Weisstein, Aug 31 2003

Corrected by Martin Fuller, Jul 12 2007

A074454 Consider 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 resulting volume.

Original entry on oeis.org

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

Offset: 1

Robert G. Wilson v, Aug 22 2002

The dimension is given in A074455.

If you set v(n) = Pi^(n/2)/(n/2)! and s(n) = n*Pi^(n/2)/(n/2)! and then plot {6.283*v(n-2), s(n)} for 0<=n<=20, the two curves are almost identical.

			5.277768021113400997282145864172846387529999284510173567761637340214864\
12730547017110062048407258401284645...

David Wells, The Penguin Dictionary of Curious and Interesting Numbers. Penguin Books, NY, 1986, Revised edition 1987. See p. 67.

Cf. A072345, A072346.

d = x /. FindRoot[ PolyGamma[1 + x/2] == Log[Pi], {x, 5}, WorkingPrecision -> 105]; First[ RealDigits[ Pi^(d/2)/(d/2)!]][[1 ;; 99]] (* Jean-François Alcover, Apr 12 2013 *)

Checked by Martin Fuller, Jul 12 2007

A074456 Consider surface area 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 resulting surface area.

Original entry on oeis.org

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

Offset: 2

Robert G. Wilson v, Aug 22 2002

If you set v[n_] := Pi^(n/2)/(n/2)! and s[n_] := n*Pi^(n/2)/(n/2)! and then Plot[{6.283v[n - 2], s[n]}, {n, 0, 20}], the two curves are almost identical.

			33.1611944849620026918630240155829735800472328410872...

The dimension is given in A074455.

Cf. A072478, A072479, A074454.

area[d_] := d * Pi^(d/2)/Gamma[d/2 + 1]; area[x /. FindRoot[PolyGamma[x/2] == Log[Pi], {x, 7}, WorkingPrecision -> 120]] (* Amiram Eldar, Jun 08 2023 *)

Checked by Martin Fuller, Jul 12 2007

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

A275162 Decimal expansion of dimension d in which a ball of radius 1/2 has maximum volume.

Original entry on oeis.org

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

Offset: 0

Eric R. Carter, Nov 13 2016

The definition of hypervolume for a ball of radius r, generalized to continuous dimension d, is given by ((Pi^(d/2))*(r^d))/Gamma((d/2) + 1). Assigning r = 1/2, the d > 0 which maximizes this formula is the non-integral real number 0.4765825... whose digits form this sequence.

			d = 0.47658258230608529520761576885882324030164...

Cf. A074455.

RealDigits[d/.FindRoot[Log[4/Pi] + PolyGamma[0, 1 + d/2], {d, 1}, WorkingPrecision -> 200]][[1]]

Maximizing ((Pi^(d/2))*((1/2)^d))/Gamma((d/2) + 1) for d>0 we obtain a volume of 1.0386933280526... when d equals the positive real root of the derivative: ((2^(-1-d))*(Pi^(d/2))*((log(4*Pi) + PolyGamma(0, 1+d/2))))/(Gamma(1+d/2)). - Corrected by Eric R. Carter, May 09 2019

cp's OEIS Frontend

A087300 Duplicate of A074455.

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A074457 Consider surface area 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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A074454 Consider 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 resulting volume.

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A074456 Consider surface area 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 resulting surface area.

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

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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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PARI

A275162 Decimal expansion of dimension d in which a ball of radius 1/2 has maximum volume.

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Mathematica

Formula