A102753 -id:A102753 - cp's OEIS frontend

A152584 Decimal expansion of (Pi^3)/24.

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

Offset: 1

Eric Desbiaux, Dec 08 2008

Consider infinite sum made of areas of circles Pi*radius^2 with diameter 1/n.
The volume is (Pi/4)*(1 + 1/4 + 1/9 + 1/16 + 1/25 + ... + 1/n^2)
= (1 - 1/3 + 1/5 - 1/7 + 1/9 - 1/11 + ...)*(1 + 1/4 + 1/9 + 1/16 + 1/25 + ...)
= (Pi/4) * (Pi^2/6) = Pi^3/24.
Equals volume of a cone of height Pi^2/8 and radius 1.
Equals volume of a sphere (4*Pi*Pi^2/32)/3 with radius^3 = (Pi^2/32).

			1.291928195012492507311513127795891466759387023578...

G. C. Greubel, Table of n, a(n) for n = 1..5000
Index entries for transcendental numbers

Cf. A013661, A003881, A102753, A019679, A111003.

RealDigits[(Pi^3)/24, 10, 50][[1]] (* G. C. Greubel, Jun 09 2017 *)

PARI
```
(Pi^3)/24 \\ G. C. Greubel, Jun 09 2017
```

Equals Integral_{x=0..oo} arctan(x)^2/(x^2 + 1) dx. - Amiram Eldar, Aug 06 2020

A164105 Decimal expansion of Pi^3/6.

Original entry on oeis.org

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

Offset: 1

R. J. Mathar, Aug 10 2009

Volume of the 6-dimensional unit sphere.

			Equals 5.1677127800499700292460525111835658670375480943...

G. C. Greubel, Table of n, a(n) for n = 1..10000
Eric Weisstein, Hypersphere, Wolfram Mathworld.
Wikipedia, Hypersphere
Index entries for transcendental numbers

Cf. A000796, A019699, A102753, A164103, A164106, A164108.

RealDigits[Pi^3/6, 10, 100][[1]] (* G. C. Greubel, Apr 09 2017 *)

PARI
```
Pi^3/6 \\ G. C. Greubel, Apr 09 2017
```

Equals A091925/6 = A019670*A102753.

A276023 Decimal expansion of 32*Pi^4/945.

Original entry on oeis.org

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

Offset: 1

Ilya Gutkovskiy, Aug 16 2016

Volume of the 9-dimensional unit sphere.

More generally, the ordinary generation function for the volume of the n-dimensional unit sphere is exp(Pi*x^2)*(erf(sqrt(Pi)*x) + 1) = 1 + 2*x + Pi*x^2 + (4*Pi/3)*x^3 + (Pi^2/2)*x^4 + ...

			3.2985089027387068693821065037445117...

G. C. Greubel, Table of n, a(n) for n = 1..10000
Ilya Gutkovskiy, Volume of the n-dimensional unit sphere
Eric Weisstein's World of Mathematics, Hypersphere
Index entries for transcendental numbers

Cf. A021949, A092425.

Cf. similar sequences of the volume of the n-dimensional unit sphere: A000796 (n = 2), 10*A019699 (n = 3), A102753 (n = 4), A164103 (n = 5), A164105 (n = 6), A164106 (n = 7), A164108 (n = 8).

Mathematica
```
RealDigits[(32 Pi^4)/945, 10, 120][[1]]
```

(32*Pi^4)/945 \\ G. C. Greubel, Apr 09 2017

Equals 32*A092425*A021949.

A344964 Decimal expansion of the sum of the reciprocals of the squares of the zeros of the digamma function.

Original entry on oeis.org

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

Offset: 1

Amiram Eldar, Jun 03 2021

The sum is Sum_{k>=0} 1/x_k^2, where x_k is the k-th zero of the digamma function, i.e., root of psi(x) = 0: x_0 = 1.461632... (A030169) is the only positive root, x_1 = -0.504083... (A175472), etc.

			5.26798012435239798373562163629331979431626684387002...

István Mező and Michael E. Hoffman, Zeros of the digamma function and its Barnes G-function analogue, Integral Transforms and Special Functions, Vol. 28, No. 11 (2017), pp. 846-858.
Wikipedia, Digamma function.

Cf. A344965, A344966, A344967, A344968.
Cf. A001620, A102753.
Cf. A030169, A175472, A175473, A175474, A256681, A256682, A256683, A256684, A256685, A256686, A256687.

RealDigits[Pi^2/2 + EulerGamma^2, 10, 100][[1]]

Equals Pi^2/2 + gamma^2 = A102753 + A155969, where gamma is Euler's constant (A001620).

A344965 Decimal expansion of the sum of the reciprocals of the cubes of the zeros of the digamma function (negated).

Original entry on oeis.org

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

Offset: 1

Amiram Eldar, Jun 03 2021

The sum is Sum_{k>=0} 1/x_k^3, where x_k is the k-th zero of the digamma function, i.e., root of psi(x) = 0: x_0 = 1.461632... (A030169) is the only positive root, x_1 = -0.504083... (A175472), etc.

			-7.84898826280450630489883732716055067110164127911638...

István Mező and Michael E. Hoffman, Zeros of the digamma function and its Barnes G-function analogue, Integral Transforms and Special Functions, Vol. 28, No. 11 (2017), pp. 846-858.
Wikipedia, Digamma function.

Cf. A344964, A344966, A344967, A344968.
Cf. A001620, A002117, A102753.
Cf. A030169, A175472, A175473, A175474, A256681, A256682, A256683, A256684, A256685, A256686, A256687.

RealDigits[EulerGamma*Pi^2/2 + 4*Zeta[3]  + EulerGamma^3, 10, 100][[1]]

Equals -gamma*Pi^2/2 - 4*zeta(3) - gamma^3, where gamma is Euler's constant (A001620).

A248359 Least number k such that cos(Pi/k) + 1/(k*n) > 1.

Original entry on oeis.org

5, 10, 15, 20, 25, 30, 35, 40, 45, 50, 55, 60, 65, 70, 75, 79, 84, 89, 94, 99, 104, 109, 114, 119, 124, 129, 134, 139, 144, 149, 153, 158, 163, 168, 173, 178, 183, 188, 193, 198, 203, 208, 213, 218, 223, 227, 232, 237, 242, 247, 252, 257, 262, 267, 272, 277

Offset: 1

Clark Kimberling, Oct 07 2014

It appears that a(n+1) - a(n) is in {4,5} for n >= 1.

Lim_{n->infinity} a(n)/n = Pi^2/2 = 4.9348022..., but lim_{n->infinity} (a(n+1) - a(n)) does not exist; Pi^2/2 is only a mean value of these differences. - Vaclav Kotesovec, Oct 09 2014

			Taking n = 2, we have cos(Pi/9) + 1/(18) = 0.99524... < 1 < 1.0010565... = cos(Pi/10) + 1/(20), so that a(2) = 10, as corroborated for n = 2 in the following list of approximations:
n ... cos(Pi/a(n)) + 1/(n*a(n))
1 ... 1.009016994
2 ... 1.001056516
3 ... 1.000369823
4 ... 1.000188341
5 ... 1.000114701
6 ... 1.000077451

Clark Kimberling and Vaclav Kotesovec, Table of n, a(n) for n = 1..10000 (first 500 terms from Clark Kimberling)

Cf. A102753, A248360.

z = 800; f[n_] := f[n] = Select[Range[z], Cos[Pi/#] + 1/(#*n) > 1 &, 1];
u = Flatten[Table[f[n], {n, 1, z}]]  (* A248359 *)
Table[Floor[1/(1 - Cos[Pi/n])], {n, 1, z/10}]  (* A248360 *)
Table[k=1; While[Cos[Pi/k]+1/(k*n)<=1,k++]; k,{n,1,100}] (* Vaclav Kotesovec, Oct 09 2014 *)

a(n) ~ n*Pi^2/2 = n*A102753. - Vaclav Kotesovec, Oct 09 2014

A222128 Decimal expansion of the real part of 1/i^Pi, where i=sqrt(-1).

Original entry on oeis.org

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

Offset: 0

Bruno Berselli, Feb 08 2013

Also, decimal expansion of the real part of i^Pi.

			0.22058404074969808866894591325578751045884803815941067237004887322...

Vincenzo Librandi, Table of n, a(n) for n = 0..5000

Cf. A102753, A211883, A222129 (imaginary part of 1/i^Pi).

RealDigits[Re[1/I^Pi], 10, 90][[1]] (* or *) RealDigits[Cos[Pi^2/2], 10, 90][[1]]

fpprec:90; ev(bfloat(realpart(1/%i^%pi)));

Equals cos(Pi^2/2).

A222129 Decimal expansion of the imaginary part of 1/i^Pi, where i=sqrt(-1).

Original entry on oeis.org

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

Offset: 0

Bruno Berselli, Feb 08 2013

Also, decimal expansion of the imaginary part of -i^Pi.

			0.97536797208363138515748287410849478847409651236377497298708899116...

Vincenzo Librandi, Table of n, a(n) for n = 0..5000

Cf. A102753, A211884, A222128 (real part of 1/i^Pi).

RealDigits[Im[1/I^Pi], 10, 90][[1]] (* or *) RealDigits[-Sin[Pi^2/2], 10, 90][[1]]

fpprec:90; ev(bfloat(imagpart(1/%i^%pi)));

Equals -sin(Pi^2/2).

A363876 Decimal expansion of the geometric mean of the isoperimetric quotient of ellipses when expressed in terms of their eccentricity.

Original entry on oeis.org

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

Offset: 0

Tian Vlasic, Jun 25 2023

The isoperimetric quotient of a curve is defined as Q = (4*Pi*A)/p^2, where A and p are the area and the perimeter of that curve respectively.

The isoperimetric quotient of an ellipse depends only on its eccentricity e in accordance to the formula Q = (Pi^2*sqrt(1-e^2))/(4*E(e)^2), where E() is the complete elliptic integral of the second kind.

			0.916816923382168248...

Eric Weisstein's World of Mathematics, Isoperimetric Quotient
Wikipedia, Elliptic integral

Cf. A102753, A363848, A363874.

First[RealDigits[Pi^2/2*Exp[-1 - 2*NIntegrate[Log[EllipticE[x^2]], {x, 0, 1}, WorkingPrecision -> 100]]]]

Equals ((Pi^2)/2) * exp(-1-2*Integral_{x=0..1} log(E(x)) dx).

A175295 Decimal expansion of the integral of cos(Pix)log(x)/x^2 from x=1 to infinity.

Original entry on oeis.org

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

Offset: 0

R. J. Mathar, Mar 24 2010

			0.02991320398393497843930179...

G. C. Greubel, Table of n, a(n) for n = 0..10000
R. J. Mathar, Numerical evaluation of the oscillatory integral over exp(i*pi*x)*x^(1/x) between 1 and infinity, arXiv:0912.3844 [math.CA], 2009-2010, App. B.

evalf(1+Pi^2/2*( gamma+log(Pi)-1 ) -Pi^2*hypergeom([1/2,1/2,1], [3/2,3/2,3/2,2],-Pi^2/4)/2 ) ;

Join[{0}, RealDigits[ N[1/2*(Pi^2*(-2*HypergeometricPFQ[{1/2, 1/2}, {3/2, 3/2, 3/2}, -Pi^2/4] + Log[Pi] + EulerGamma - 1) + 2*Pi*SinIntegral[Pi] - 2), 105]][[1]]] (* Jean-François Alcover, Nov 08 2012 *)
Join[{0},RealDigits[NIntegrate[Cos[Pi*x] Log[x]/x^2,{x,1,\[Infinity]}, WorkingPrecision->1000],10,120][[1]]] (* Harvey P. Dale, Nov 01 2017 *)

1+ A102753*( A053510 -1 + A001620 - 3F4(1/2,1/2,1; 3/2,3/2,3/2,2 ; -A091476) ) .

cp's OEIS Frontend

A152584 Decimal expansion of (Pi^3)/24.

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A164105 Decimal expansion of Pi^3/6.

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A276023 Decimal expansion of 32*Pi^4/945.

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A344964 Decimal expansion of the sum of the reciprocals of the squares of the zeros of the digamma function.

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A344965 Decimal expansion of the sum of the reciprocals of the cubes of the zeros of the digamma function (negated).

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A248359 Least number k such that cos(Pi/k) + 1/(k*n) > 1.

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A222128 Decimal expansion of the real part of 1/i^Pi, where i=sqrt(-1).

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A175295 Decimal expansion of the integral of cos(Pix)log(x)/x^2 from x=1 to infinity.