A053510 -id:A053510 - cp's OEIS frontend

A061444 Decimal expansion of log(2 * Pi).

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

Offset: 1

Frank Ellermann, Jun 11 2001

Used in formulas for gamma(x), e.g., in Stirling's approximation for m!.
Also decimal expansion of zeta'(0)/zeta(0). - Benoit Cloitre, Sep 28 2002
The value of log(2*Pi) is close to 1 + Sum_{n>=2} log(zeta(n)) = 1.83067035427178011248.... - Arkadiusz Wesolowski, Jul 17 2011

			1.837877066409345483560659472811235279722794947275566825634303...

Harry J. Smith, Table of n, a(n) for n = 1..20000
Simon Plouffe, log(2*Pi) to 10000 digits
Simon Plouffe, Log(2*pi) to 2000 places

Cf. A002162, A053510, A094642, A131659.

RealDigits[N[Log[2*Pi], 100]][[1]] (* Arkadiusz Wesolowski, Aug 29 2011 *)

{ default(realprecision, 20080); x=log(2*Pi); for (n=1, 20000, d=floor(x); x=(x-d)*10; write("b061444.txt", n, " ", d)) } \\ Harry J. Smith, Jul 22 2009

Equals A002162 + A053510 = A131659 - A094642. - R. J. Mathar, Aug 27 2011

Equals 1 + Sum_{k>=1} zeta(2*k)/(k*(2*k + 1)). - Amiram Eldar, Aug 20 2020

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.

A094642 Decimal expansion of log(Pi/2).

Original entry on oeis.org

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

Offset: 0

Jonathan Sondow and Robert G. Wilson v, May 18 2004

			log(Pi/2) = 0.45158270528945486472619522989488214357179467855505...

George Boros and Victor Moll, Irresistible Integrals: Symbolics, Analysis and Experiments in the Evaluation of Integrals, Cambridge University Press, Cambridge, 2004, Chap. 7.
Jonathan Borwein and Peter Borwein, Pi and the AGM, John Wiley & Sons, New York, 1987, Chap. 11.
Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Section 1.6.3, pp. 43-44.

Dirk Huylebrouck, Similarities in irrationality proofs for Pi, ln2, zeta(2) and zeta(3), Amer. Math. Monthly, Vol. 108, No. 3 (2001), pp. 222-231.
Jonathan Sondow, A faster product for pi and a new integral for ln(pi/2), The American Mathematical Monthly, Vol. 112, No. 8 (2005), pp. 729-734; Editor's endnotes, ibid., Vol. 113, No. 7 (2006), pp. 670-671; arXiv preprint, arXiv:math/0401406 [math.NT], 2004.

Cf. A019669, A094643.

Mathematica
```
RealDigits[ Log[Pi/2], 10, 111][[1]]
```

log(Pi/2) \\ Charles R Greathouse IV, Jun 23 2014

Equals Sum_{n>=1} zeta(2*n)/(n*2^(2*n)) (cf. Boros & Moll p. 131). - Jean-François Alcover, Apr 29 2013
Equals Re(log(log(I))). - Stanislav Sykora, May 09 2015
Equals Integral_{-oo..+oo} -log(1/2 + i*z)/cosh(Pi*z) dz, where i is the imaginary unit. - Peter Luschny, Apr 08 2018
Equals Integral_{0..Pi/2} (2/(Pi-2*t)-tan(t)) dt. - Clark Kimberling, Jul 10 2020
Equals -Sum_{k>=1} log(1 - 1/(2*k)^2). - Amiram Eldar, Aug 12 2020
Equals Sum_{k>=1} (-1)^(k+1) * log(1 + 1/k). - Amiram Eldar, Jun 26 2021
Equals A053510 - A002162. - R. J. Mathar, Jun 15 2023

A053511 Decimal expansion of log_10 (Pi).

Original entry on oeis.org

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

Offset: 0

Hsu, Po-Wei (Benny) (arsene_lupin(AT)intekom.co.za), Jan 14 2000

			0.49714987269413385435126828829089887365167832438...

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

Cf. A053510 (log(Pi)), A002392 (log(10)).

SetDefaultRealField(RealField(100)); R:= RealField(); Log(10, Pi(R)); // G. C. Greubel, May 15 2019

RealDigits[Log[10, Pi], 10, 105][[1]] (* Alonso del Arte, Sep 05 2012 *)

log(Pi)/log(10) \\ Charles R Greathouse IV, Sep 05 2012

numerical_approx(log(pi, 10), digits=100) # G. C. Greubel, May 15 2019

More terms from James Sellers, Jan 20 2000

A115252 Decimal expansion of -(Pilog((sqrt(2Pi)*Gamma(3/4))/Gamma(1/4)))/2.

Original entry on oeis.org

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

Offset: 0

Eric W. Weisstein, Jan 17 2006

This sequence (its negated version) is also the decimal expansion of the first Malmsten integral int_{x=1..infinity} log(log(x))/(1 + x^2) dx = int_{x=0..1} log(log(1/x))/(1 + x^2) dx = int_{x=0..infinity} 0.5*log(x)/cosh(x) dx = int_{x=Pi/4..Pi/2} log(log(tan(x))) dx = (1/2)*Pi*log(2) + (3/4)*Pi*log(Pi) - Pi*log(Gamma(1/4)). - Iaroslav V. Blagouchine, Mar 29 2015

			0.26044280630098844554009386878972721953181917772314...

G. C. Greubel, Table of n, a(n) for n = 0..5000
I. V. Blagouchine, Rediscovery of Malmsten's integrals, their evaluation by contour integration methods and some related results, The Ramanujan Journal, Volume 35, Issue 1, pp. 21-110, 2014, DOI: 10.1007/s11139-013-9528-5. PDF file
Eric Weisstein's World of Mathematics, Vardi's Integral

Cf. A256127 (second Malmsten integral), A256128 (third Malmsten integral), A256129 (fourth Malmsten integral), A068466 (Gamma(1/4)), A256166 (log(Gamma(1/4))), A002162 (log 2), A053510 (log Pi).

RealDigits[-Pi/2*Log[Sqrt[2 Pi] Gamma[3/4]/Gamma[1/4]], 10, 111][[1]] (* Robert G. Wilson v, Dec 06 2014 *)

(-Pi*log((sqrt(2*Pi)*gamma(3/4))/gamma(1/4)))/2 \\ Michel Marcus, Dec 06 2014

Equals integral_[0..1] log(1/log(1/x))/(1+x^2) dx. - Jean-François Alcover, Jan 28 2015

A059561 Beatty sequence for log(Pi).

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 8, 9, 10, 11, 12, 13, 14, 16, 17, 18, 19, 20, 21, 22, 24, 25, 26, 27, 28, 29, 30, 32, 33, 34, 35, 36, 37, 38, 40, 41, 42, 43, 44, 45, 46, 48, 49, 50, 51, 52, 53, 54, 56, 57, 58, 59, 60, 61, 62, 64, 65, 66, 67, 68, 69, 70, 72, 73, 74, 75, 76, 77, 78, 80, 81

Offset: 1

Mitch Harris, Jan 22 2001

a(n) is the largest integer m such that e^m < Pi^n. - Stanislav Sykora, May 29 2015

Harry J. Smith, Table of n, a(n) for n = 1..2000
Aviezri S. Fraenkel, Jonathan Levitt and Michael Shimshoni, Characterization of the set of values f(n)=[n alpha], n=1,2,..., Discrete Math. 2 (1972), no.4, 335-345.
Eric Weisstein's World of Mathematics, Beatty Sequence.
Index entries for sequences related to Beatty sequences

Beatty complement is A059562.
Cf. A000796 (Pi), A001113 (e), A053510 (log(Pi)).
Cf. A022932 (characteristic function).

Floor[Range[100]*Log[Pi]] (* Paolo Xausa, Jul 05 2024 *)

{ default(realprecision, 100); b=log(Pi); for (n = 1, 2000, write("b059561.txt", n, " ", floor(n*b)); ) } \\ Harry J. Smith, Jun 28 2009

a(n) = A004777(n+1), 1 <= n < 83. - R. J. Mathar, Oct 05 2008

a(n) = floor(n*log(Pi)). - Michel Marcus, Jan 04 2015

A231736 Decimal expansion of the natural logarithm of Pi^Pi.

Original entry on oeis.org

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

Offset: 1

Stanislav Sykora, Nov 13 2013

			3.59627499972915819808600175164636038136917928975387723049724412082...

Stanislav Sykora, Table of n, a(n) for n = 1..5000

Cf. A000796, A053510, A073233, A231737.

RealDigits[Pi * Log[Pi], 10, 120][[1]] (* Amiram Eldar, May 17 2023 *)

PARI
```
Pi*log(Pi)
```

Equals Pi*log(Pi).

A256128 Decimal expansion of the third Malmsten integral: int_{x=1..infinity} log(log(x))/(1 - x + x^2) dx, negated.

Original entry on oeis.org

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

Offset: 0

Iaroslav V. Blagouchine, Mar 15 2015

			-0.671719601885874542354405069288779884008802066219356...

G. C. Greubel, Table of n, a(n) for n = 0..5000
Iaroslav V. Blagouchine, Rediscovery of Malmsten's integrals, their evaluation by contour integration methods and some related results, The Ramanujan Journal, Volume 35, Issue 1, pp. 21-110, 2014, DOI: 10.1007/s11139-013-9528-5. PDF file
Wikipedia, Carl Malmsten

Cf. A115252 (first Malmsten integral), A256127 (second Malmsten integral), A256129 (fourth Malmsten integral), A073005 (Gamma(1/3)), A002162 (log 2), A002391 (log 3), A053510 (log Pi), A002194 (sqrt 3).

evalf(Pi*(7*log(2)+8*log(Pi)-3*log(3)-12*log(GAMMA(1/3)))/(3*sqrt(3)),120); # Vaclav Kotesovec, Mar 17 2015

RealDigits[Pi*(7*Log[2]+8*Log[Pi]-3*Log[3]-12*Log[Gamma[1/3]])/(3*Sqrt[3]),10,105][[1]] (* Vaclav Kotesovec, Mar 17 2015 *)

Pi*(7*log(2)+8*log(Pi)-3*log(3)-12*log(gamma(1/3)))/(3*sqrt(3)) \\ Michel Marcus, Mar 18 2015

Equals integral_{x=0..1} log(log(1/x))/(1 - x + x^2) dx.
Equals integral_{x=0..infinity} log(x)/(1 - 2*cosh(x)) dx.
Equals Pi*(7*log(2) + 8*log(Pi) - 3*log(3) - 12*log(Gamma(1/3)))/(3*sqrt(3)).

A231737 Decimal expansion of the natural logarithm of Pi^(1/Pi).

Original entry on oeis.org

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

Offset: 0

Stanislav Sykora, Nov 13 2013

			0.3643788396759062570495877303161624138917070390986055504746692...

Stanislav Sykora, Table of n, a(n) for n = 0..5000

Cf. A000796, A053510, A073238, A231736.

RealDigits[Log[Pi]/Pi, 10, 120][[1]] (* Amiram Eldar, May 17 2023 *)

PARI
```
log(Pi)/Pi
```

Equals log(Pi)/Pi.

A256129 Decimal expansion of the fourth Malmsten integral: int_{x=1..infinity} log(log(x))/(1 + x)^2 dx, negated.

Original entry on oeis.org

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

Offset: 0

Iaroslav V. Blagouchine, Mar 15 2015

			-0.0628164798060389979401584300937601437351823286924336...

G. C. Greubel, Table of n, a(n) for n = 0..5000
Iaroslav V. Blagouchine, Rediscovery of Malmsten's integrals, their evaluation by contour integration methods and some related results, The Ramanujan Journal, Volume 35, Issue 1, pp. 21-110, 2014, DOI: 10.1007/s11139-013-9528-5. PDF file
Wikipedia, Carl Malmsten

A115252 (first Malmsten integral), A256127 (second Malmsten integral), A256128 (third Malmsten integral), A002162 (log 2), A053510 (log Pi), A001620 (Euler's constant, gamma).

evalf((log(Pi/2)-gamma)/2,120); # Vaclav Kotesovec, Mar 17 2015

RealDigits[(Log[Pi/2]-EulerGamma)/2,10,105][[1]] (* Vaclav Kotesovec, Mar 17 2015 *)

(-Euler+log(Pi)-log(2))/2 \\ Michel Marcus, Mar 18 2015

Equals integral_{x=0..1} log(log(1/x))/(1 + x)^2 dx.
Equals integral_{x=0..infinity} 0.5*log(x)/(1 + cosh(x)) dx.
Equals (log(Pi) - log(2) - gamma)/2.

cp's OEIS Frontend

A061444 Decimal expansion of log(2 * Pi).

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

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A094642 Decimal expansion of log(Pi/2).

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A053511 Decimal expansion of log_10 (Pi).

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Magma

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A115252 Decimal expansion of -(Pi*log((sqrt(2*Pi)*Gamma(3/4))/Gamma(1/4)))/2.

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A059561 Beatty sequence for log(Pi).

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A231736 Decimal expansion of the natural logarithm of Pi^Pi.

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A115252 Decimal expansion of -(Pilog((sqrt(2Pi)*Gamma(3/4))/Gamma(1/4)))/2.