A091925 -id:A091925 - cp's OEIS frontend

A164106 Decimal expansion of 16*Pi^3/105.

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

Offset: 1

R. J. Mathar, Aug 10 2009

Volume of the 7-dimensional unit sphere.

			Equals 4.72476597033140116959639086736783164986290111...

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, A164105, A164108.

RealDigits[16*Pi^3/105, 10, 100][[1]] (* G. C. Greubel, Apr 09 2017 *)

16*Pi^3/105 \\ G. C. Greubel, Apr 09 2017

Equals 16*A091925/105 = A164107/7 .

A308637 Decimal expansion of Pi^3/Zeta(3).

Original entry on oeis.org

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

Offset: 2

Seiichi Manyama, Aug 23 2019

Index entries for zeta function.

-----+---------------------------------
   n |  Zeta(n)
-----+---------------------------------
   2 |     Pi^2  / 6         = A013661.
   3 |     Pi^3  / 25.79...  = A002117.
   4 |     Pi^4  / 90        = A013662.
   5 |     Pi^5  / A309926   = A013663.
   6 |     Pi^6  / 945       = A013664.
   7 |     Pi^7  / A309927   = A013665.
   8 |     Pi^8  / 9450      = A013666.
   9 |     Pi^9  / A309928   = A013667.
  10 |     Pi^10 / 93555     = A013668.
  11 |     Pi^11 / A309929   = A013669.
  12 | 691*Pi^12 / 638512875 = A013670.
...
Cf. A002432, A091925, A276120 (Zeta(3)/Pi^3).

RealDigits[Pi^3/Zeta[3], 10, 100][[1]] (* Amiram Eldar, Aug 24 2019 *)

PARI
```
Pi^3/zeta(3)
```

Pi^3/Zeta(3) = A091925/A002117.

More terms from Amiram Eldar, Aug 24 2019

A164104 Decimal expansion of 8*Pi^2/3.

Original entry on oeis.org

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

Offset: 2

R. J. Mathar, Aug 10 2009

Surface area of the 5-dimensional unit sphere.

			Equals 26.318945069571622983558642666336...

G. C. Greubel, Table of n, a(n) for n = 2..5000
Eric Weisstein, Hypersphere, Wolfram Mathworld.
Wikipedia, Hypersphere
Index entries for transcendental numbers

Cf. A019692, A019694, A164102, A091925, A164107, A164109.

RealDigits[8*Pi^2/3, 10, 50][[1]] (* G. C. Greubel, Sep 11 2017 *)

PARI
```
8*Pi^2/3 \\ G. C. Greubel, Sep 11 2017
```

Equals 5*A164103 = 10 * A019699 * A019692.

A-number in formula corrected by R. J. Mathar, Aug 12 2010

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.

A164107 Decimal expansion of 16*Pi^3/15.

Original entry on oeis.org

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

Offset: 2

R. J. Mathar, Aug 10 2009

Surface area of the 7-dimensional unit sphere.

			Equals 33.07336179231980818717473607157482154904030780...

G. C. Greubel, Table of n, a(n) for n = 2..5000
Eric Weisstein, Hypersphere, Wolfram Mathworld.
Wikipedia, Hypersphere
Index entries for transcendental numbers

Equals 7*A164106.

Cf. A019692, A019694, A164102, A164104, A091925, A164109.

RealDigits[16*Pi^3/15, 10, 50][[1]] (* G. C. Greubel, Sep 11 2017 *)

16*Pi^3/15 \\ G. C. Greubel, Sep 11 2017

A212003 Decimal expansion of (2*Pi)^3.

Original entry on oeis.org

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

Offset: 3

Omar E. Pol, Aug 11 2012

			248.05021344239856140381052053681116161780230852708...

Juan M. Maldacena, The Large N Limit of Superconformal Field Theories and Supergravity, arXiv:hep-th/9711200v3, 1997-1998, p. 5.
Index entries for transcendental numbers

Cf. A000796, A019692, A091925, A212002, A212004.

RealDigits[(2*Pi)^3,10,120][[1]] (* Harvey P. Dale, Oct 09 2013 *)

8*Pi^3 \\ Charles R Greathouse IV, Oct 02 2022

Equals Product_{k=1..14, gcd(k,14)==1} Gamma(k/14). - Amiram Eldar, Jun 12 2021

A100322 a(n) is the smallest positive integer k such that the digits of the fractional part of Pi^k begin with n.

Original entry on oeis.org

1, 7, 6, 4, 8, 23, 25, 2, 15, 91, 51, 307, 49, 1, 102, 315, 112, 12, 76, 26, 115, 208, 77, 276, 161, 40, 13, 41, 7, 99, 174, 169, 86, 453, 110, 204, 53, 6, 67, 4, 228, 123, 37, 134, 158, 192, 33, 45, 61, 200, 31, 324, 8, 56, 34, 105, 148, 17, 19, 92, 23, 38, 27, 39, 32, 82

Offset: 1

Mark Hudson (mrmarkhudson(AT)hotmail.com), Nov 16 2004

			Pi^1 = 3.14159..., whose digits after the decimal point begin with 1, so a(1)=1.
Pi^2 = 9.869..., whose digits after the decimal point begin with 8, so a(8)=2.
a(14)=1 because Pi^1 = 3.14....

Cf. A000796, A002388, A091925, A092425, A092731, A092732, A092735, A092736.

a(n) = my(k=1); while (floor(frac(Pi^k)*10^(1+logint(n, 10))) != n, k++); k; \\ Michel Marcus, Jun 18 2022

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

A193716 Decimal expansion of Pi^3log(2)/24 - 3Pi*zeta(3)/16.

Original entry on oeis.org

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

Offset: 0

Seiichi Kirikami, Aug 03 2011

The absolute value of the integral {x=0..Pi/2} x^2*log(sin(x )) dx or (d^2/da^2 (integral {x=0..Pi/2} cos(ax)*log(sin(x )) dx)) at a=0. The absolute value of (sum {n=1..infinity} (limit { a -> 0} (d^2/da^2 (sin((a+2n)*Pi/2)/n/(a+2n)))))-(Pi/2)^3*log(2)/3. [Seiichi Kirikami and Peter J. C. Moses]

			0.18742642282823108026...

I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, series and Products, 1.441.2, 4th edition, log(sin(x))=-(sum {1..infinity} cos(2nx)/n)-log(2).

R. E. Crandall, J. P. Buhler, On the evaluation of Euler sums, Exper. Math. 3 (4) (1994) 275 (discuss int_{0..1} x^n*cot(x) dx which is obtained by partial integration).
S. Koyama and N. Kurokawa, Euler’s integrals and multiple sine functions, Proc. Amer. Math. Soc. 133(2005), 1257-1265.

Cf. A173623, A173624, A193717.

RealDigits[ N[Pi (2 Pi^2 Log[2] - 9 Zeta[3]) / 48, 105] ][[1]]

Pi^3*log(2)/24 - 3*Pi*zeta(3)/16 \\ Michel Marcus, Oct 25 2017

Equals A091925*A002162/24-3*A000796*A002117/16.

cp's OEIS Frontend

A164106 Decimal expansion of 16*Pi^3/105.

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A308637 Decimal expansion of Pi^3/Zeta(3).

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A164104 Decimal expansion of 8*Pi^2/3.

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A164105 Decimal expansion of Pi^3/6.

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A164107 Decimal expansion of 16*Pi^3/15.

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A212003 Decimal expansion of (2*Pi)^3.

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A100322 a(n) is the smallest positive integer k such that the digits of the fractional part of Pi^k begin with n.

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

A193716 Decimal expansion of Pi^3log(2)/24 - 3Pi*zeta(3)/16.