A094642 -id:A094642 - 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

A257817 Decimal expansion of the real part of li(i), i being the imaginary unit.

Original entry on oeis.org

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

Offset: 0

Stanislav Sykora, May 10 2015

li(x) is the logarithmic integral function, extended to the whole complex plane. The corresponding imaginary part is in A257818.

			0.47200065143956865077760610761412783650733054301836188186838371...

Stanislav Sykora, Table of n, a(n) for n = 0..2000
Eric Weisstein's World of Mathematics, Logarithmic Integral
Wikipedia, Logarithmic integral function

Cf. A001620, A019669, A094642, A257818.

evalf(Re(Li(I)),120); # Vaclav Kotesovec, May 10 2015

RealDigits[Re[LogIntegral[I]], 10, 120][[1]] (* Vaclav Kotesovec, May 10 2015 *)

li(z) = {my(c=z+0.0*I); \\ If z is real, convert it to complex
  if(imag(c)<0, return(-Pi*I-eint1(-log(c))),
  return(+Pi*I-eint1(-log(c)))); }
  a=real(li(I))

Equals gamma + log(Pi/2) + Sum_{k>=1}((-1)^k*(Pi/2)^(2*k)/(2*k)!/(2*k)).

Equals Ci(Pi/2), the maximum value of the cosine integral along the real axis. - Stanislav Sykora, Nov 12 2016

A256358 Decimal expansion of log(sqrt(Pi/2)).

Original entry on oeis.org

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

Offset: 0

Jean-François Alcover, Mar 26 2015

Equals the derivative of the Dirichlet eta function at x=0. - Stanislav Sykora, May 27 2015

			0.22579135264472743236309761494744107178589733927752815869647153...

G. C. Greubel, Table of n, a(n) for n = 0..5000
J.-F. Alcover, Plot of harmonic sum G(x) for x >= 0 .
Henri Cohen, Fernando Rodriguez Villegas, Don Zagier, Convergence Acceleration of Alternating Series, Exp. Math. 9 (1) (2000) 3-12.
Costas Efthimiou, Problem 1838, Mathematics Magazine, Vol. 83, No. 1 (2010), p. 65; A weakly convergent series of logs, Solution to Problem 1838 by Tiberiu Trif, ibid., Vol. 84, No. 1 (2011), pp. 65-67.
Philippe Flajolet, Xavier Gourdon, and Philippe Dumas, Mellin Transforms and Asymptotics: Harmonic Sums.
Eric Weisstein's World of Mathematics, Dirichlet eta function.

Cf. A069998, A094642.

RealDigits[Log[Sqrt[Pi/2]], 10, 105] // First
RealDigits[DirichletEta'[0], 10, 110][[1]] (* Eric W. Weisstein, Jan 06 2024 *)

log(sqrt(Pi/2)) \\ G. C. Greubel, Jan 09 2017

Given the harmonic sum G(x) = Sum_{k>=1} (-1)^k*log(k)*exp(-k^2*x), lim_{x->0} G(x) = log(sqrt(Pi/2)).
Integral_{x=0..oo} G(x) dx = (Pi^2/12)*log(2) + zeta'(2)/2 = (Pi^2/12)*(EulerGamma + log(4*Pi) - 12*log(Glaisher)) = 0.1013165781635...
G'(0) = 7*zeta'(-2) = -7*zeta(3)/(4*Pi^2) = -0.2131391994...
Equals Integral_{-oo..+oo} -log(1/2 + i*z)/(exp(-Pi*z) + exp(Pi*z)) dz, where i is the imaginary unit. - Peter Luschny, Apr 08 2018
Equals Sum_{n>=0} Sum_{m>=1} (-1)^(m+n) * log(m+n)/(m+n) (Efthimiou, 2010). - Amiram Eldar, Apr 09 2022
Equals A094642/2. - R. J. Mathar, Jun 15 2023

A216582 Decimal expansion of the logarithm of Pi to base 2.

Original entry on oeis.org

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

Offset: 1

Alonso del Arte, Sep 09 2012

			1.651496129472318798...

Jörg Arndt and Christoph Haenel, Pi Unleashed, New York: Springer (2001), p. 239.

G. C. Greubel, Table of n, a(n) for n = 1..10000

Cf. A002162, A020857, A053510, A053511, A061444, A094642.

SetDefaultRealField(RealField(100)); R:= RealField(); Log(Pi(R))/Log(2); // G. C. Greubel, Apr 08 2019

evalf(log[2](Pi)) ; # R. J. Mathar, Sep 11 2012

Mathematica
```
RealDigits[Log[2, Pi], 10, 105][[1]]
```

log(Pi)/log(2) \\ Michel Marcus, Mar 05 2019

log(pi,2).n(digits=100) # Jani Melik, Oct 05 2012

Log_2(Pi) = log(Pi) / log(2) = A053510 / A002162.

Equals (A061444 / A002162) - 1 = (A094642 / A002162) + 1. - John W. Nicholson, Mar 12 2019

A377588 Decimal expansion of 7zeta(3)/(2Pi^2) - log(2) + 1/2.

Original entry on oeis.org

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

Offset: 0

Stefano Spezia, Nov 02 2024

			0.233131218257560481506289305137990304982506635269...

Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Section 1.6.3, pp. 43-44.

Cf. A002117, A002162, A094642, A164102, A244009, A377589, A377592.

RealDigits[7Zeta[3]/(2Pi^2)-Log[2]+1/2,10,100][[1]]

Equals Sum_{k>=1} zeta(2*k)/((k + 1)*2^(2*k)) (see Finch).

A377589 Decimal expansion of 9zeta(3)/(2Pi^2) - log(2) + 1/3.

Original entry on oeis.org

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

Offset: 0

Stefano Spezia, Nov 02 2024

			0.18825837982446689796062876035594274490384190278...

Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Section 1.6.3, pp. 43-44.

Cf. A002117, A002162, A094642, A164102, A244009, A377588, A377592.

RealDigits[9Zeta[3]/(2Pi^2)-Log[2]+1/3,10,100][[1]]

Equals Sum_{k>=1} zeta(2*k)/((2*k + 3)*2^(2*k-1)) (see Finch).

A377592 Decimal expansion of 9zeta(3)/Pi^2 - 93zeta(5)/(2*Pi^4) - log(2) + 1/4.

Original entry on oeis.org

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

Offset: 0

Stefano Spezia, Nov 02 2024

			0.158000963625557733268629385978458549091780284796...

Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Section 1.6.3, pp. 43-44.

Cf. A002117, A002162, A002388, A013663, A092425, A094642, A244009, A377588, A377589.

RealDigits[9Zeta[3]/Pi^2-93Zeta[5]/(2Pi^4)-Log[2]+1/4,10,100][[1]]

Equals Sum_{k>=1} zeta(2*k)/((k + 2)*2^(2*k)) (see Finch).

A068449 Factorial expansion of log(Pi/2) = sum n>0 a(n)/n!.

Original entry on oeis.org

0, 0, 2, 2, 4, 1, 0, 7, 7, 3, 3, 6, 4, 10, 9, 1, 15, 2, 8, 10, 14, 6, 4, 7, 3, 3, 2, 2, 7, 26, 3, 30, 3, 31, 9, 29, 23, 12, 29, 3, 0, 12, 1, 11, 4, 13, 22, 17, 24, 33, 40, 34, 48, 27, 15, 5, 33, 33, 51, 48, 42, 46, 47, 38, 35, 30, 27, 1, 51, 52, 28, 25, 13, 30, 51, 14, 39, 12, 9, 58, 33

Offset: 1

Benoit Cloitre, Mar 10 2002

G. C. Greubel, Table of n, a(n) for n = 1..10000

Cf. A007514, A094642 (decimal expansion).

R:= RealField(); [Floor(Log(Pi(R)/2))] cat [Floor(Factorial(n)*Log(Pi(R)/2)) - n*Floor(Factorial((n-1))* Log(Pi(R)/2)) : n in [2..30]]; // G. C. Greubel, Mar 21 2018

Table[If[n == 1, Floor[Log[Pi/2]], Floor[n!*Log[Pi/2]] - n*Floor[(n - 1)!*Log[Pi/2]]], {n, 1, 50}] (* G. C. Greubel, Mar 21 2018 *)

for(n=1,30, print1(if(n==1, floor(log(Pi/2)), floor(n!*log(Pi/2)) - n*floor((n-1)!*log(Pi/2))), ", ")) \\ G. C. Greubel, Mar 21 2018

A094643 Continued fraction for log(Pi/2).

Original entry on oeis.org

0, 2, 4, 1, 1, 1, 33, 1, 4, 2, 1, 2, 1, 17, 1, 1, 4, 4, 1, 2, 1, 3, 1, 3, 1, 17, 54, 1, 4, 1, 3, 38, 1, 2, 1, 1, 2, 3, 4, 3, 1, 4, 1, 8, 4, 2, 1, 4, 12, 1, 1, 1, 2, 1, 1, 1, 3, 1, 1, 1, 1, 1, 2, 1, 1, 16, 3, 2, 4, 1, 5, 1, 12, 1, 2, 14, 1, 1, 1, 2, 3, 2, 16, 3, 4, 4, 1, 1, 10, 198, 2, 6, 2, 1, 2, 3, 1, 2

Offset: 0

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

G. Boros and V. Moll, Irresistible Integrals: Symbolics, Analysis and Experiments in the Evaluation of Integrals, Cambridge University Press, Cambridge, 2004, Chap. 7.
J. Borwein and P. Borwein, Pi and the AGM, John Wiley & Sons, New York, 1987, Chap. 11.

D. Huylebrouck, Similarities in irrationality proofs for Pi, ln2, zeta(2) and zeta(3), Amer. Math. Monthly 108 (2001) 222-231.
Jonathan Sondow, A faster product for pi and a new integral for ln(pi/2), arXiv:math/0401406 [math.NT], 2004; Amer. Math. Monthly 112 (2005), 729-734 and 113 (2006), 670.

Cf. A094642 (decimal expansion).

Mathematica
```
ContinuedFraction[ Log[Pi/2], 100]
```

Offset changed by Andrew Howroyd, Aug 07 2024

A255681 Decimal expansion of Sum_{k>=1} zeta(2k+1)/((2k+1)2^(2k)).

Original entry on oeis.org

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

Offset: 0

Jean-François Alcover, Apr 13 2015

			0.1159315156584124488107200313757741370333407984203316553149...

G. C. Greubel, Table of n, a(n) for n = 0..10000
H. M. Srivastava and Junesang Choi, Zeta and q-Zeta Functions and Associated Series and Integrals, Elsevier Insights, 2011, pp. 272 and 314.
Eric Weisstein's MathWorld, Riemann Zeta Function.
Wikipedia, Riemann Zeta Function.

Cf. A001620, A002162, A094642 (= log(Pi/2) = Sum_{k>=2} zeta(2*k)/(k*2^(2*k))).

SetDefaultRealField(RealField(100)); R:= RealField(); Log(2) - EulerGamma(R); // G. C. Greubel, Sep 06 2018

RealDigits[Log[2] - EulerGamma, 10, 105] // First

default(realprecision, 100); log(2) - Euler \\ G. C. Greubel, Sep 06 2018

Equals log(2) - EulerGamma.
Equals Sum_{k>=1} (zeta(2*k+1)-1)/(k+1). - Amiram Eldar, May 24 2021
Equals Sum_{k>=1} psi(k)/2^k, where psi(x) is the digamma function. - Amiram Eldar, Sep 12 2022

cp's OEIS Frontend

A061444 Decimal expansion of log(2 * Pi).

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A257817 Decimal expansion of the real part of li(i), i being the imaginary unit.

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A256358 Decimal expansion of log(sqrt(Pi/2)).

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A216582 Decimal expansion of the logarithm of Pi to base 2.

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A377588 Decimal expansion of 7*zeta(3)/(2*Pi^2) - log(2) + 1/2.

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A377589 Decimal expansion of 9*zeta(3)/(2*Pi^2) - log(2) + 1/3.

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A377592 Decimal expansion of 9*zeta(3)/Pi^2 - 93*zeta(5)/(2*Pi^4) - log(2) + 1/4.

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A377588 Decimal expansion of 7zeta(3)/(2Pi^2) - log(2) + 1/2.

A377589 Decimal expansion of 9zeta(3)/(2Pi^2) - log(2) + 1/3.

A377592 Decimal expansion of 9zeta(3)/Pi^2 - 93zeta(5)/(2*Pi^4) - log(2) + 1/4.

A255681 Decimal expansion of Sum_{k>=1} zeta(2k+1)/((2k+1)2^(2k)).