A115252 -id:A115252 - cp's OEIS frontend

A256166 Decimal expansion of log(Gamma(1/4)).

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

Offset: 1

Iaroslav V. Blagouchine, Mar 17 2015

			1.288022524698077457370610440219717295925377565112860...

G. C. Greubel, Table of n, a(n) for n = 1..5000

Cf. A068466 (Gamma(1/4)), A115252 (first Malmsten's integral).

Cf. decimal expansions of log(Gamma(1/k)): A155968 (k=2), A256165 (k=3), A256167 (k=5), A255888 (k=6), A256609 (k=7), A255306 (k=8), A256610 (k=9), A256612 (k=10), A256611 (k=11), A256066 (k=12), A256614 (k=16), A256615 (k=24), A256616 (k=48).

evalf(log(GAMMA(1/4)),120); # Vaclav Kotesovec, Mar 17 2015

RealDigits[Log[Gamma[1/4]],10,105][[1]] (* Vaclav Kotesovec, Mar 17 2015 *)

log(gamma(1/4)) \\ Michel Marcus, Mar 17 2015

A256127 Decimal expansion of the second Malmsten integral: Integer_{x >= 1} log(log(x))/(1 + x + x^2) dx, negated.

Original entry on oeis.org

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

Offset: 0

Iaroslav V. Blagouchine, Mar 15 2015

			-0.12632148170620903636522675325320239184424430946528...

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), A256128 (third Malmsten integral) , A256129 (fourth Malmsten integral), A073005 (Gamma(1/3)), A256165 (log(Gamma(1/3))), A061444 (log(2*Pi)), A002391 (log 3), A002194 (sqrt 3).

evalf(Pi*(8*log(2*Pi) - 3*log(3) - 12*log(GAMMA(1/3)))/(6*sqrt(3)),120); # Vaclav Kotesovec, Mar 17 2015

RealDigits[Integrate[Log[Log[1/x]]/(1 + x + x^2), {x, 0, 1}], 10, 100][[1]] (* Alonso del Arte, Mar 16 2015 *)
RealDigits[Pi*(8*Log[2*Pi] - 3*Log[3] - 12*Log[Gamma[1/3]])/(6*Sqrt[3]),10,105][[1]] (* Vaclav Kotesovec, Mar 17 2015 *)

Pi*(8*log(2*Pi) - 3*log(3) - 12*log(gamma(1/3)))/(6*sqrt(3)) \\ Michel Marcus, Mar 18 2015

intnum(x=0, 1, log(log(1/x))/(1 + x + x^2))

intnum(x=1, oo, log(log(x))/(1 + x + x^2))

intnum(x=0, [oo, 1], log(x)/(1 + 2*cosh(x))) \\ Gheorghe Coserea, Sep 26 2018

Equals Integral_{x=0..1} log(log(1/x))/(1 + x + x^2) dx.
Equals Integral_{x>=0} log(x)/(1 + 2*cosh(x)) dx.
Equals Pi*(8*log(2*Pi) - 3*log(3) - 12*log(Gamma(1/3)))/(6*sqrt(3)).

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)).

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.

A241017 Decimal expansion of Sierpiński's S constant, which appears in a series involving the function r(n), defined as the number of representations of the positive integer n as a sum of two squares. This S constant is the usual Sierpiński K constant divided by Pi.

Original entry on oeis.org

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

Offset: 0

Jean-François Alcover, Aug 08 2014

			0.822825249678847032995328716261464949475693118894850218393815613...

Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 2.10 Sierpinski's Constant, p. 123.

M. W. Coffey, Summatory relations and prime products for the Stieltjes constants, and other related results, arXiv:1701.07064 (2017) Proposition 9.
Steven R. Finch, Mathematical Constants II, Encyclopedia of Mathematics and Its Applications, Cambridge University Press, Cambridge, 2018, p. 103.
Guillaume Melquiond, W. Georg Nowak, Paul Zimmermann, Numerical approximation of the Masser-Gramain constant to four decimal places, Mathematics of Computation, Volume 82, Number 282, April 2013, Pages 1235-1246
Eric Weisstein's MathWorld, Sierpiński's Constant
Wikipedia, Sierpiński's Constant

Cf. A062089, A062539, A068466, A086058.

S = Log[4*Pi^3*Exp[2*EulerGamma]/Gamma[1/4]^4]; RealDigits[S, 10, 104] // First

log(agm(sqrt(2), 1)^2/2) + 2*Euler \\ Charles R Greathouse IV, Nov 26 2024

S = gamma + beta'(1) / beta(1), where beta is Dirichlet's beta function.
S = log(Pi^2*exp(2*gamma) / (2*L^2)), where L is Gauss' lemniscate constant.
S = log(4*Pi^3*exp(2*gamma) / Gamma(1/4)^4), where gamma is Euler's constant and Gamma is Euler's Gamma function.
S = A062089 / Pi, where A062089 is Sierpiński's K constant.
S = A086058 - 1, where A086058 is the conjectured (but erroneous!) value of Masser-Gramain 'delta' constant. [updated by Vaclav Kotesovec, Apr 27 2015]
S = 2*gamma + (4/Pi)*integral_{x>0} exp(-x)*log(x)/(1-exp(-2*x)) dx.
Sum_{k=1..n} r(k)/k = Pi*(log(n) + S) + O(n^(-1/2)).
Equals 2*A001620 - A088538*A115252 [Coffey]. - R. J. Mathar, Jan 15 2021

A257406 Decimal expansion of Integral_{0..infinity} log(x)/cosh(x) dx (negated).

Original entry on oeis.org

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

Offset: 0

Jean-François Alcover, Apr 22 2015

			-0.5208856126019768910801877375794544390636383554462853499753755842...

G. C. Greubel, Table of n, a(n) for n = 0..5000
Victor Adamchik, A Class of Logarithmic Integrals (1997) Proc. of ISSAC'97

Cf. A068466, A115252.

RealDigits[(Pi/2)*Log[4*Pi^3/Gamma[1/4]^4], 10, 103] // First
RealDigits[Integrate[-Log[x]/Cosh[x],{x,0,\[Infinity]}],10,120][[1]] (* Harvey P. Dale, Feb 05 2025 *)

(Pi/2)*log(4*Pi^3/gamma(1/4)^4) \\ Michel Marcus, Apr 22 2015

(Pi/2)*log(4*Pi^3/Gamma(1/4)^4).
Also equals 2*Integral_{0..1} (1/(x^2+1))*log(log(1/x)) dx.
Also equals 2*Integral_{Pi/4..Pi/2} log(log(tan(x))) dx.

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A256166 Decimal expansion of log(Gamma(1/4)).

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A256127 Decimal expansion of the second Malmsten integral: Integer_{x >= 1} log(log(x))/(1 + x + x^2) dx, negated.

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

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

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A241017 Decimal expansion of Sierpiński's S constant, which appears in a series involving the function r(n), defined as the number of representations of the positive integer n as a sum of two squares. This S constant is the usual Sierpiński K constant divided by Pi.

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A257406 Decimal expansion of Integral_{0..infinity} log(x)/cosh(x) dx (negated).

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