A002161 -id:A002161 - cp's OEIS frontend

A092731 Decimal expansion of Pi^5.

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

Offset: 3

Mohammad K. Azarian, Apr 12 2004

			306.0196847852814532

G. C. Greubel, Table of n, a(n) for n = 3..10000
Jean-Christophe Pain, The fifth power of Pi: new series representation involving the golden ratio and an application in physics, 2022.
Jean-Christophe Pain, New series representations for any positive power of Pi from a relation involving trigonometric functions, 2208.02624 [math.NT], 2022.
Kh. Hessami Pilehrood and Tatiana Hessami Pilehrood, Series acceleration formulas for beta values, Discr. Math. Theor. Comp. Sci. 12 (2) (2010) 223-236.
Index entries for transcendental numbers

Cf. A000796, A002161, A019692, A091925, A092735, A000364, A002437.

R:= RealField(100); (Pi(R))^5; // G. C. Greubel, Mar 09 2018

RealDigits[Pi^5, 10, 100][[1]] (* G. C. Greubel, Mar 09 2018 *)

PARI
```
Pi^5 \\ G. C. Greubel, Mar 09 2018
```

From Peter Bala, Oct 31 2019: (Start)
Pi^5 = (4!/(2*305)) * Sum_{n >= 0} (-1)^n*( 1/(n + 1/6)^5 + 1/(n + 5/6)^5 ), where 305 = ((3^5 + 1)/4)*A000364(2) = A002437(2).
Pi^5 = (4!/(2*3905)) * Sum_{n >= 0} (-1)^n*( 1/(n + 1/10)^5 - 1/(n + 3/10)^5 - 1/(n + 7/10)^5 + 1/(n + 9/10)^5 ), where 3905 = ((5^5 - 1)/4)*A000364(2).
Cf. A019692, A091925 and A092735. (End)

A020759 Decimal expansion of (-1)*Gamma'(1/2)/Gamma(1/2) where Gamma(x) denotes the Gamma function.

Original entry on oeis.org

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

Offset: 1

Benoit Cloitre, May 24 2003

Decimal expansion of -psi(1/2). - Benoit Cloitre, Mar 07 2004

			1.96351002602142347944097633299875556719315960466...

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), 6.3.3, p. 258. - Robert G. Wilson v, Jun 20 2011
S. J. Patterson, An introduction to the theory of the Riemann zeta function, Cambridge studies in advanced mathematics no. 14, p. 135.

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Wikipedia, Digamma function.
Index entries for sequences related to the digamma function.

Cf. A001620, A002161, A002162, A131265, A248176.

R:=RealField(100); EulerGamma(R) + 2*Log(2); // G. C. Greubel, Aug 27 2018

evalf(-Psi(0.5)) ; # R. J. Mathar, Sep 10 2013

RealDigits[ EulerGamma + 2 Log[2], 10, 111][[1]] (* Robert G. Wilson v, Jun 20 2011 *)

PARI
```
Euler+2*log(2)
```

2-psi(-1/2) \\ Stanislav Sykora, Oct 03 2014

Gamma'(1/2)/Gamma(1/2) = -EulerGamma - 2*log(2) = -1.9635100260214234794... where EulerGamma is the Euler-Mascheroni constant (A001620).
Equals 2 - psi(-1/2) = 2-A248176. - Stanislav Sykora, Oct 03 2014
Equals A131265/A002161. - R. J. Mathar, Jun 02 2022
Equals lim_{n->oo} (Sum_{k=0..n} 1/(k+1/2) - log(n)). - Amiram Eldar, Mar 04 2023

A155968 Decimal expansion of (1/2)*log(Pi).

Original entry on oeis.org

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

Offset: 0

R. J. Mathar, Jan 31 2009

This sequence is also the decimal expansion of the logarithm of the Gamma-function at 1/2. - Iaroslav V. Blagouchine, Mar 20 2015

			0.572364942924700087071713675676529355823...

G. C. Greubel, Table of n, a(n) for n = 0..5000

Cf. A053510.

Maple
```
evalf(log(Pi)/2);
```

RealDigits[Log[Pi]/2,10,120][[1]] (* Harvey P. Dale, May 31 2015 *)

log(gamma(1/2)) \\ or \\ log(Pi)/2 \\ G. C. Greubel, Jan 16 2017

Equals A053510/2 = log(A002161) = A131659/4.

A175379 Decimal expansion of Gamma(1/6).

Original entry on oeis.org

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

Offset: 1

R. J. Mathar, Apr 24 2010

A175379 * A073005 * A002161 * A073006 * A203145 = 4*sqrt(Pi^5/3), which is the case n=6 of Product_{i=1..n-1} Gamma(i/n) = sqrt((2*Pi)^(n-1)/n). - Bruno Berselli, Dec 18 2012

The transcendence of this constant is in the mathematical folklore; see Finch (who credits Nesterenko) and Gun-Murty-Rath. - Charles R Greathouse IV, Nov 11 2013

			Equals 5.56631600178023...

G. C. Greubel, Table of n, a(n) for n = 1..5000
Steven Finch, Errata and addenda to Mathematical Constants (2013)
Sanoli Gun, M. Ram Murty, and Purusottam Rath, Transcendence of the log gamma function and some discrete periods, J. Number Theory (2009), doi:10.1016/j.jnt.2009.01.008
R. Vidunas, Expressions for values of the Gamma function, arxiv:math/0403510 [math.CA], 2004.
Wikipedia, Particular values of the Gamma function: General rational arguments
Index entries for transcendental numbers
Index entries to sequences related to the Gamma function

Cf. A073006, A203145.

SetDefaultRealField(RealField(100)); Gamma(1/6); // G. C. Greubel, Mar 10 2018

Maple
```
evalf(GAMMA(1/6)) ;
```

RealDigits[Gamma[1/6], 10, 110][[1]] (* Bruno Berselli, Dec 13 2012 *)

gamma(1/6) \\ Charles R Greathouse IV, Nov 16 2013

Equals 2*Pi/A203145 = A002194 * A073005^2 / (A002161 * A002580) = A019692 / 1.12878703....

A092732 Decimal expansion of Pi^6.

Original entry on oeis.org

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

Offset: 3

Mohammad K. Azarian, Apr 12 2004

			961.389193575304437...

G. C. Greubel, Table of n, a(n) for n = 3..10000
Mohammad Reza Yegan, On the irrationality of Pi^4 and Pi^6, Journal of Number Theory, Volume 178, September 2017, Pages 5-10.
Index entries for transcendental numbers

Cf. A000796, A002161.

R:= RealField(100); (Pi(R))^6; // G. C. Greubel, Mar 09 2018

RealDigits[Pi^6, 10, 108][[1]] (* Alonso del Arte, Oct 05 2014 *)

PARI
```
Pi^6 \\ G. C. Greubel, Mar 09 2018
```

Equals Sum_{k>=1} k*(k+1)*(k+2)*(k+3)*(k+4)*zeta(k+5)/2^(k+2). - Amiram Eldar, May 21 2021

A087016 Decimal expansion of G(3/2), where G is the Barnes G-function.

Original entry on oeis.org

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

Offset: 1

Eric W. Weisstein, Jul 30 2003

			1.0692...

Eric Weisstein's World of Mathematics, Barnes G-Function
Wikipedia, Barnes G-function

Cf. A087013, A087014, A087015, A087017.

(2^(1/24)*E^(1/8)*Pi^(1/4))/Glaisher^(3/2)
(* Or, since version 7.0, *) RealDigits[BarnesG[3/2], 10, 102] // First (* Jean-François Alcover, Jul 11 2014 *)

2^(1/24)*Pi^(1/4)*exp(3/2*zeta'(-1)) \\ Charles R Greathouse IV, Dec 12 2013

Equals A087014 * A002161. - R. J. Mathar, Jul 24 2025

A203145 Decimal expansion of Gamma(5/6).

Original entry on oeis.org

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

Offset: 1

N. J. A. Sloane, Dec 29 2011

			1.1287870299081259612609010902588420133267874416647554517520...

Cf. A002161, A019692, A073005, A073006, A175379.

RealDigits[Gamma[5/6], 10, 100][[1]] (* Bruno Berselli, Dec 18 2012 *)

gamma(5/6) \\ Charles R Greathouse IV, Nov 26 2024

A073005 * this * A231863 * A010768 = A073006. - R. J. Mathar, Jan 15 2021
Equals 2*Pi/Gamma(1/6) = A019692 / A175379. - Amiram Eldar, Jul 04 2023
Equals 2^(4/3) * Pi^(3/2) / (sqrt(3) * Gamma(1/3)^2). - Vaclav Kotesovec, Jul 04 2023

A257955 Decimal expansion of Gamma(1/Pi).

Original entry on oeis.org

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

Offset: 1

Iaroslav V. Blagouchine, May 14 2015

The reference gives an interesting product representation in terms of rational multiple of 1/Pi for Gamma(1/Pi).

			2.8112975146708616421227908037104816935281655223291765...

G. C. Greubel, Table of n, a(n) for n = 1..5000
Iaroslav V. Blagouchine, Two series expansions for the logarithm of the gamma function involving Stirling numbers and containing only rational coefficients for certain arguments related to 1/Pi, Mathematics of Computation (AMS), 2015.

Cf. A257957, A257958, A257959, A002161, A073005, A068466, A175380, A175379, A220086, A203142, A256190, A256191, A256192, A203140, A203139, A203138, A203137.

Maple
```
evalf(GAMMA(1/Pi), 117);
```
Mathematica
```
RealDigits[Gamma[1/Pi], 10, 117][[1]]
```

default(realprecision, 117); gamma(1/Pi)

A092735 Decimal expansion of Pi^7.

Original entry on oeis.org

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

Offset: 4

Mohammad K. Azarian, Apr 12 2004

Wentworth (1903) shows how to compute the tangent of 15 degrees (A019913) to five decimal places by the laborious process of adding up the first few terms of Pi/12 + Pi^3/5184 + 2Pi^5/3732480 + 17Pi^7/11287019520 + ... - Alonso del Arte, Mar 13 2015

			3020.293227776792067514206493...

George Albert Wentworth, New Plane and Spherical Trigonometry, Surveying, and Navigation. Boston: The Atheneum Press (1903): 240.

G. C. Greubel, Table of n, a(n) for n = 4..10000
Index entries for transcendental numbers

Cf. A000796, A002161, A019692, A091925, A092731, A000364, A002437.

R:= RealField(100); (Pi(R))^7; // G. C. Greubel, Mar 09 2018

RealDigits[Pi^7, 10, 100][[1]] (* Alonso del Arte, Mar 13 2015 *)

PARI
```
Pi^7 \\ G. C. Greubel, Mar 09 2018
```

From Peter Bala, Oct 30 2019: (Start)
Pi^7 = (6!/(2*33367)) * Sum_{n >= 0} (-1)^n*( 1/(n + 1/6)^7 + 1/(n + 5/6)^7 ), where 33367 = ((3^7 + 1)/4)*A000364(3) = A002437(3).
Pi^7 = (6!/(2*1191391)) * Sum_{n >= 0} (-1)^n*( 1/(n + 1/10)^7 - 1/(n + 3/10)^7 - 1/(n + 7/10)^7 + 1/(n + 9/10)^7 ), where 1191391 = ((5^7 - 1)/4)*A000364(3). (End)

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A019645 Decimal expansion of sqrt(Pi*e).

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A092731 Decimal expansion of Pi^5.

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A020759 Decimal expansion of (-1)*Gamma'(1/2)/Gamma(1/2) where Gamma(x) denotes the Gamma function.

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A155968 Decimal expansion of (1/2)*log(Pi).

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A175379 Decimal expansion of Gamma(1/6).

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A092732 Decimal expansion of Pi^6.

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