A002391 -id:A002391 - cp's OEIS frontend

A256843 Decimal expansion of the generalized Euler constant gamma(2,3).

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

Offset: 0

Jean-François Alcover, Apr 11 2015

			0.07320737570615959366903185990752913907462383026830934562939...

G. C. Greubel, Table of n, a(n) for n = 0..10000
D. H. Lehmer, Euler constants for arithmetic progressions, Acta Arith. 27 (1975), p. 134.

Cf. A001620 (gamma(1,1) = EulerGamma), A002391, A200064.
Primitive ruler-and-compass constructible gamma(r,k): A228725 (1,2), A256425 (1,3), A256778 (1,4), A256779 (1,5), A256780 (2,5), A256781 (1,8), A256782 (3,8), A256783 (1,12), A256784 (5,12).
Other gamma(r,k) (1 <= r <= k <= 5): A239097 (2,2),  A256843 (2,3),  A256844 (3,3), A256845 (2,4), A256846 (3,4), A256847 (4,4), A256848 (3,5), A256849 (4,5), A256850 (5,5).

SetDefaultRealField(RealField(100)); R:= RealField(); EulerGamma(R)/3 - Pi(R)/(6*Sqrt(3)) + Log(3)/6; // G. C. Greubel, Aug 28 2018

Join[{0}, RealDigits[-Log[3]/3 - PolyGamma[2/3]/3, 10, 105] // First]

default(realprecision, 100); Euler/3 - Pi/(6*sqrt(3)) + log(3)/6 \\ G. C. Greubel, Aug 28 2018

Equals EulerGamma/3 - Pi/(6*sqrt(3)) + log(3)/6.

Equals -(psi(2/3) + log(3))/3 = (A200064 - A002391)/3. - Amiram Eldar, Jan 07 2024

A016578 Decimal expansion of log(3/2).

Original entry on oeis.org

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

Offset: 0

N. J. A. Sloane

			0.4054651081081643819780131154643491365719904234624941976140143...

L. B. W. Jolley, Summation of Series, Dover (1961), eq (102), page 20.

Harry J. Smith, Table of n, a(n) for n = 0..20000
Index entries for transcendental numbers

Cf. A016529, A002391, A002162, A083679.

RealDigits[Log[3/2],10,111][[1]] (* Robert G. Wilson v, Aug 08 2011 *)

default(realprecision, 20080); x=10*log(3/2); for (n=0, 20000, d=floor(x); x=(x-d)*10; write("b016578.txt", n, " ", d)); \\ Harry J. Smith, May 17 2009

Equals Sum {k>=1} 1/(k*3^k). - Robert G. Wilson v, Aug 08 2011
Equals 1/2 - 1/(2*2^2) + 1/(3*2^3) - 1/(4*2^4) + ... [Jolley].
Equals A002391-A002162. - Michel Marcus, Sep 17 2016
From Amiram Eldar, Aug 07 2020: (Start)
Equals 2 * arctanh(1/5).
Equals Integral_{x=0..oo} 1/(2*exp(x) + 1) dx. (End)
log(3/2) = 2*Sum_{n >= 1} 1/(n*P(n, 5)*P(n-1, 5)), where P(n, x) denotes the n-th Legendre polynomial. The first 10 terms of the series gives the approximation log(3/2) = 0.40546510810816438197(04...), correct to 20 decimal places. - Peter Bala, Mar 16 2024
Equals Sum_{n >= 1} (-1)^(n+1) * 5/(n*binomial(2*n, n)*6^n). The n-th term of the series is O(5*sqrt(Pi/n)*1/24^n). - Peter Bala, Mar 04 2025
Equals Integral_{x=0..1} (sqrt(x) - 1)/log(x) dx. - Kritsada Moomuang, Jun 14 2025

A304656 Decimal expansion of Pi*sqrt(3).

Original entry on oeis.org

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

Offset: 1

Peter Luschny, May 16 2018

			5.4413980927026535517822347729264671968521987442782217267096548061643695433790...

C. Elsner, On recurrence formulas for sums involving binomial coefficients, Fib. Q., 43,1 (2005), 31-45.
Melissa Larson, Verifying and discovering BBP-type formulas, 2008.
Wikipedia, Bailey-Borwein-Plouffe formula.
Index entries for transcendental numbers

Cf. A002388, A002391, A093602, A248682.

Maple
```
Pi*sqrt(3): evalf(%, 100);
```

RealDigits[N[StieltjesGamma[0,1/6]-StieltjesGamma[0,5/6],99]][[1]] (* corrected by Harvey P. Dale, Oct 13 2020 *)
RealDigits[Pi Sqrt[3],10,120][[1]] (* Harvey P. Dale, Oct 13 2020 *)

# Use several guard digits when computing.
# BBP formula (9/32) P(1, 64, 6, (16, 8, 0, -2, -1, 0)).
from decimal import Decimal as dec, getcontext
def BBPpisqrt3(n: int) -> dec:
    getcontext().prec = n
    s = dec(0); f = dec(1); g = dec(64)
    for k in range(int(n * 0.5536546824812272) + 1):
        sixk = dec(6 * k)
        s += f * ( dec(16) / (sixk + 1) + dec(8) / (sixk + 2)
                 - dec(2)  / (sixk + 4) - dec(1) / (sixk + 5) )
        f /= g
    return (s * dec(9)) / dec(32)
print(BBPpisqrt3(200))  # Peter Luschny, Nov 03 2023

Equals gamma(0, 1/6) - gamma(0, 5/6) where gamma(n,x) denotes the generalized Stieltjes constants.
Equals PolyGamma[0, 5/6] - PolyGamma[0, 1/6].
Equals 3*sqrt(2*zeta(2)).
Pi^2 = A304656 * A093602.
From Amiram Eldar, Aug 06 2020: (Start)
Equals Sum_{k>=0} 1/((k + 1/3)*(k + 2/3)).
Equals Integral_{x=0..oo} log(1 + 3/x^2) dx. (End)
Equals (27*S - 36)/8, where S = A248682. - Peter Luschny, Jul 22 2022
From Peter Bala, Oct 26 2023: (Start)
sqrt(3)*Pi = 9/2 + 9*Sum_{n >= 1} (-1)^(n+1)/(9*n^2 - 1);
sqrt(3)*Pi = 5 + 10*Sum_{n >= 1} 1/((4*n^2 - 1)*(9*n^2 - 1)) = 43/8 + 8*Sum_{n >= 2} (-1)^n/((n^2 - 1)*(9*n^2 - 1));
sqrt(3)*Pi = 1765/324 - (80/9)*Sum_{n >= 2} 1/((n^2 - 1)*(4*n^2 - 1)*(9*n^2 - 1)).
The following two series representations for the constant
sqrt(3)*Pi = 72 * Sum_{n >= 0} (2*n + 1)/((6*n + 1)*(6*n + 3)*(6*n + 5)) and
sqrt(3)*Pi = 8192/1485 - 860160 * Sum_{n >= 0} (2*n + 3)/((6*n + 1)*(6*n + 3)*...*(6*n + 17)) appear to generalize as follows:
for k >= 0, sqrt(3)*Pi = c(k) + (-1)^k*d(k)*Sum_{n >= 0} (2*n + 2*k + 1)/((6*n + 1)*(6*n + 3)*...*(6*n + 12*k + 5)), where c(k) is a rational number approximating sqrt(3)*Pi and d(k) = (6*k + 1)! * 2^(6*k+3) / 3^(3*k-2).
The first few values of c(k) for k >= 0 are [0, 8192/1485, 11341398016/2085060285, 62809601736704/11542783997745, 889063287831973723111424/ 163388820474305231710905, ...].
The following two series representations for the constant
sqrt(3)*Pi = 256/45 - 2560*Sum_{n >= 0} 1/((6*n + 1)*(6*n + 3)*...*(6*n + 11)) and
sqrt(3)*Pi = 337117184/62026965 + 2018508800*Sum_{n >= 0} 1/((6*n + 1)*(6*n + 3)*...*(6*n + 23)) appear to generalize as follows:
for k >= 0, sqrt(3)*Pi = c(k) - (-1)^k*d(k)*Sum_{n >= 0} 1/((6*n + 1)*(6*n + 3)*...*(6*n + 12*k + 11)), where c(k) is a rational number approximating sqrt(3)*Pi and d(k) = (6*k + 5)! * 2^(6*k+6) / 3^(3*k+1).
The first few values of c(k) for k >= 0 are [256/45, 337117184/62026965, 1732370763874304/318357429615225, 733187044080753836032/134742553582636674675, 6361250411469779336874164224/1169047010493653932891525275, ...]. (End)
For arbitrary integer k, Pi*sqrt(3) = Sum_{n >= 0} (1/(n - k + 1/6) - 1/(n + k + 5/6)) = Sum_{n >= 0} (1/(n + k + 7/6) - 1/(n - k - 1/6)). - Peter Bala, Jul 10 2024

A016632 Decimal expansion of log(9).

Original entry on oeis.org

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

Offset: 1

N. J. A. Sloane

			2.197224577336219382790490473845051409294981115645498903469388667274988...

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 2.

Harry J. Smith, Table of n, a(n) for n = 1..20000
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
Index entries for transcendental numbers

Cf. A016737 Continued fraction.

First[RealDigits[Log[9], 10, 100]] (* Paolo Xausa, Mar 21 2024 *)

default(realprecision, 20080); x=log(9); for (n=1, 20000, d=floor(x); x=(x-d)*10; write("b016632.txt", n, " ", d)); \\ Harry J. Smith, May 16 2009

log(9) = 2*Sum_{n >= 1} 1/(n*P(n, 5/4)*P(n-1, 5/4)), where P(n, x) denotes the n-th Legendre polynomial. The first 20 terms of the series gives the approximation log(9) = 2.19722457733(34...), correct to 11 decimal places. - Peter Bala, Mar 18 2024

Equals 2*A002391. - R. J. Mathar, Jun 10 2024

A016635 Decimal expansion of log(12).

Original entry on oeis.org

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

Offset: 1

N. J. A. Sloane

			2.484906649788000310229709479838878840798490826543259959976...

Milton Abramowitz and Irene A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 2.

Harry J. Smith, Table of n, a(n) for n = 1..20000
Milton Abramowitz and Irene A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
Index entries for transcendental numbers

Cf. A016740 Continued fraction.

RealDigits[Log[12], 10, 120][[1]] (* Alonso del Arte, Mar 12 2015 *)

default(realprecision, 20080); x=log(12); for (n=1, 20000, d=floor(x); x=(x-d)*10; write("b016635.txt", n, " ", d)); \\ Harry J. Smith, May 16 2009, corrected May 19 2009

Equals 2*A002162 + A002391. - R. J. Mathar, Jun 10 2024

A154920 Denominators of a ternary BBP-type formula for log(3).

Original entry on oeis.org

1, 18, 27, 324, 405, 4374, 5103, 52488, 59049, 590490, 649539, 6377292, 6908733, 66961566, 71744535, 688747536, 731794257, 6973568802, 7360989291, 69735688020, 73222472421, 690383311398, 721764371007, 6778308875544

Offset: 0

Jaume Oliver Lafont, Jan 17 2009, Jan 18 2009

nonn

log(3) = Sum_{k>=0} (9/(2k+1)+1/(2k+2))/9^(k+1).

log(3) = 1 + Sum_{k>=0} (1/(2k+2)+1/(2k+3))/9^(k+1).

David H. Bailey, A Compendium of BBP-Type Formulas for Mathematical Constants, 2017, page 14. [From _Jaume Oliver Lafont_, Sep 25 2009]
Index entries for linear recurrences with constant coefficients, signature (0,18,0,-81).

Cf. A002391, A058962.

Cf. A164985, A165132. - Jaume Oliver Lafont, Sep 25 2009

[(2-(-1)^n)*(n+1)*3^n: n in [0..30]]; // Vincenzo Librandi, Jul 06 2015

LinearRecurrence[{0,18,0,-81},{1,18,27,324},30] (* Harvey P. Dale, Jan 10 2017 *)

a(n)=(n+1)*9^((n+1)\2) \\ Jaume Oliver Lafont, Mar 25 2009

a(n) = (n+1)*9^[(n+1)/2] = 18*a(n-2) - 81*a(n-4).
Sum_{n>=0} 1/a(n) = log(3).
G.f.: (1+18*x+9*x^2)/(1-9*x^2)^2. - Jaume Oliver Lafont, Jan 29 2009
a(n) = (2-(-1)^n)*(n+1)*3^n. - Jaume Oliver Lafont, Sep 27 2009
Sum_{n>=0} (-1)^n/a(n) = log(8/3). - Amiram Eldar, Feb 26 2022

A016650 Decimal expansion of log(27).

Original entry on oeis.org

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

Offset: 1

N. J. A. Sloane

			3.295836866004329074185735710767577113942471673468248355204083000912482... - _Harry J. Smith_, May 20 2009

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 2.

Harry J. Smith, Table of n, a(n) for n = 1..20000
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
Index entries for transcendental numbers

Cf. A016455 (continued fraction).

RealDigits[Log[27],10,120][[1]] (* Harvey P. Dale, May 07 2012 *)

default(realprecision, 20080); x=log(27); for (n=1, 20000, d=floor(x); x=(x-d)*10; write("b016650.txt", n, " ", d)); \\ Harry J. Smith, May 20 2009

Log(27) = Integral_{x = 0..oo} sin(4*x)^3/x^2 dx. - Peter Bala, Nov 04 2019
From Peter Bala, Feb 27 2024: (Start)
Equals 4 - 2*Sum_{k >= 0} 1/((2*k + 1)*(2*k + 3)*4^k).
Continued fraction: 4 - 16/(24 - 96/(84 - 1440/(186 - ... - 16*n*(n + 1)(4*n^2 - 1)/((2*(n + 1)*(10*n + 11)) - ... )))). (End)
Equals 3*A002391. - R. J. Mathar, Jul 22 2025

A097321 a(n) = (3n-1) 3n (3*n+1).

Original entry on oeis.org

24, 210, 720, 1716, 3360, 5814, 9240, 13800, 19656, 26970, 35904, 46620, 59280, 74046, 91080, 110544, 132600, 157410, 185136, 215940, 249984, 287430, 328440, 373176, 421800, 474474, 531360, 592620, 658416, 728910, 804264, 884640, 970200, 1061106, 1157520

Offset: 1

Ralf Stephan, Aug 07 2004

Vincenzo Librandi, Table of n, a(n) for n = 1..10000
S. Ramanujan, Notebook entry.
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Cf. A069072, A069140, A002391, A016767.

[27*n^3-3*n: n in [1..40]]; // Vincenzo Librandi, Sep 07 2011

Table[27n^3-3n,{n,40}]  (* Harvey P. Dale, Mar 30 2011 *)

G.f.: 6x * (4x^2 + 19x + 4) / (1-x)^4.
Sum_{n>=1} 1/a(n) = (log(3) - 1)/2. - Amiram Eldar, Jul 04 2020
Sum_{n>=1} (-1)^(n+1)/a(n) = 1/2 - 2*log(2)/3. - Amiram Eldar, May 15 2022
E.g.f.: 3*exp(x)*x*(8 + 27*x + 9*x^2). - Stefano Spezia, Feb 20 2025

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)).

cp's OEIS Frontend

A256843 Decimal expansion of the generalized Euler constant gamma(2,3).

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A016578 Decimal expansion of log(3/2).

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A304656 Decimal expansion of Pi*sqrt(3).

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A016632 Decimal expansion of log(9).

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A016635 Decimal expansion of log(12).

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A154920 Denominators of a ternary BBP-type formula for log(3).

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Formula

A016650 Decimal expansion of log(27).

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A097321 a(n) = (3n-1) 3n (3*n+1).