A002194 -id:A002194 - cp's OEIS frontend

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

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)).

A381671 Decimal expansion of the isoperimetric quotient of a regular tetrahedron.

Original entry on oeis.org

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

Offset: 0

Paolo Xausa, Mar 03 2025

Polya (1954) defines the isoperimetric quotient of a solid as 36*Pi*V^2/(S^3), where V and S are the volume and surface area of the solid, respectively.

The isoperimetric quotient of a sphere is 1.

			0.30229989403903630843234637627369262204734437468212...

George Polya, Mathematics and Plausible Reasoning, Vol. 1: Induction and Analogy in Mathematics, Princeton University Press, Princeton, New Jersey, 1954. See pp. 188-189, exercise 43.

Paolo Xausa, Table of n, a(n) for n = 0..10000

Cf. A273633 (sphericity).
Cf. isoperimetric quotient of other Platonic solids: A019673 (cube), A073010 (octahedron), A374772 (dodecahedron), A381672 (icosahedron).
Cf. A002194, A019673.

First[RealDigits[Pi/(6*Sqrt[3]), 10, 100]]

Equals Pi/(6*sqrt(3)) = A019673/A002194.

A384141 Decimal expansion of the surface area of an elongated pentagonal bipyramid with unit edge.

Original entry on oeis.org

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

Offset: 1

Paolo Xausa, May 20 2025

The elongated pentagonal bipyramid is Johnson solid J_16.

			9.3301270189221932338186158537646809173570131345...

Paolo Xausa, Table of n, a(n) for n = 1..10000
Wikipedia, Elongated pentagonal bipyramid.

Cf. A384140 (volume).
Cf. A002163.
Essentially the same as A120011.

First[RealDigits[5*(2 + Sqrt[3])/2, 10, 100]] (* or *)
First[RealDigits[PolyhedronData["J16", "SurfaceArea"], 10, 100]]

Equals 5*(2 + sqrt(3))/2 = 5*(2 + A002194)/2.
Equals the largest root of 4*x^2 - 40*x + 25.
Equals 10*A019884^2. - R. J. Mathar, Sep 05 2025

A384473 Decimal expansion of the middle interior angle (in degrees) in Albrecht Dürer's approximate construction of the regular pentagon.

Original entry on oeis.org

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

Offset: 3

Stefano Spezia, May 30 2025

			108.366120162561467008046935277164429896133431...

Alfred S. Posamentier and Herbert A. Hauptman, 101 great ideas for introducing key concepts in mathematics: a resource for secondary school teachers, Corwin Press, Inc., 2001. See pages 144-145.
Alfred S. Posamentier, Math Charmers, Tantalizing Tidbits for the Mind, Prometheus Books, NY, 2003, page 181-182.

Alexander Bogomolny, Approximate Construction of Regular Pentagon by A. Dürer.
Stefano Spezia, Albrecht Dürer's approximate construction of the regular pentagon.
Wikipedia, Albrecht Dürer.

Cf. A228719, A384474 (in radians).
Cf. A002194, A019824, A072097, A177870.
Cf. A384475, A384476, A384477, A384478.

RealDigits[(3Pi/4-ArcSin[Sqrt[3]Sin[Pi/12]])180/Pi,10,100][[1]] (* or *)
RealDigits[(Pi+ArcTan[(3-Sqrt[3]+Sqrt[6Sqrt[3]-4])/(3-Sqrt[3]-Sqrt[6Sqrt[3]-4])])180/Pi,10,100][[1]]

Equals 135 - 180*arcsin(sqrt(3)*sin(Pi/12))/Pi.
Equals (Pi + arctan((3 - sqrt(3) + sqrt(6*sqrt(3) - 4))/(3 - sqrt(3) - sqrt(6*sqrt(3) - 4))))*180/Pi.
Equals (540 - 2*A384475 - A384477)/2.
A384475 < this constant < A384477.

A384474 Decimal expansion of the middle interior angle (in radians) in Albrecht Dürer's approximate construction of the regular pentagon.

Original entry on oeis.org

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

Offset: 1

Stefano Spezia, May 30 2025

			1.891345594448510418687173478952739199024779225...

Alfred S. Posamentier and Herbert A. Hauptman, 101 great ideas for introducing key concepts in mathematics: a resource for secondary school teachers, Corwin Press, Inc., 2001. See pages 144-145.
Alfred S. Posamentier, Math Charmers, Tantalizing Tidbits for the Mind, Prometheus Books, NY, 2003, page 181-182.

Alexander Bogomolny, Approximate Construction of Regular Pentagon by A. Dürer.
Stefano Spezia, Albrecht Dürer's approximate construction of the regular pentagon.
Wikipedia, Albrecht Dürer.

Cf. A228719, A384473 (in degrees).
Cf. A002194, A019824, A177870.
Cf. A384475, A384476, A384477, A384478.

RealDigits[3Pi/4-ArcSin[Sqrt[3]Sin[Pi/12]],10,100][[1]] (* or *)
RealDigits[Pi+ArcTan[(3-Sqrt[3]+Sqrt[6Sqrt[3]-4])/(3-Sqrt[3]-Sqrt[6Sqrt[3]-4])],10,100][[1]]

Equals 3*Pi/4 - arcsin(sqrt(3)*sin(Pi/12)).
Equals Pi + arctan((3 - sqrt(3) + sqrt(6*sqrt(3) - 4))/(3 - sqrt(3) - sqrt(6*sqrt(3) - 4))).
Equals (3*Pi - 2*A384476 - A384478)/2.
A384476 < this constant < A384478.

A010504 Decimal expansion of square root of 51.

Original entry on oeis.org

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

Offset: 1

N. J. A. Sloane

Continued fraction expansion is 7 followed by {7, 14} repeated. - Harry J. Smith, Jun 06 2009

			7.141428428542849997999399811367265278766171159902733833208430882765820...

Harry J. Smith, Table of n, a(n) for n = 1..20000
Index entries for algebraic numbers, degree 2.

Cf. A040043 (continued fraction).

Digits:=100; evalf(sqrt(51)); # Wesley Ivan Hurt, Mar 04 2014

RealDigits[N[Sqrt[51],200]][[1]] (* Vladimir Joseph Stephan Orlovsky, Feb 24 2011 *)

{ default(realprecision, 20080); x=sqrt(51); for (n=1, 20000, d=floor(x); x=(x-d)*10; write("b010504.txt", n, " ", d)); } \\ Harry J. Smith, Jun 06 2009

Equals A002194*A010473. - R. J. Mathar, Jun 08 2025

A165922 Decimal expansion of 2sqrt(3)/(9Pi).

Original entry on oeis.org

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

Offset: 0

Rick L. Shepherd, Sep 30 2009

The ratio of the volume of a regular tetrahedron to the volume of the circumscribed sphere. (The MathWorld link shows that the circumradius for a tetrahedron with side length a is a*sqrt(6)/4.)

			0.122517532315953788780294777402882098...

Frank Jackson and Eric W. Weisstein, Tetrahedron
Index entries for transcendental numbers

Cf. A000796, A002194, A010469, A049541, A020784.

RealDigits[(2Sqrt[3])/(9Pi),10,120][[1]] (* Harvey P. Dale, Nov 17 2013 *)

PARI
```
2*3^(-3/2)/Pi
```

2*sqrt(3)/(9*Pi) = A010469/(9*A000796) = (2/9)*A002194/A000796 = (2/9)*A002194*A049541 = 2*A020784/A000796 = 2*3^(-3/2)/Pi.

A176394 Decimal expansion of 3+2*sqrt(3).

Original entry on oeis.org

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

Offset: 1

Klaus Brockhaus, Apr 16 2010

Continued fraction expansion of 3+2*sqrt(3) is A010696 preceded by 6.
a(n) = A010469(n) for n > 1.
Largest radius of three circles tangent to a circle of radius 1. - Charles R Greathouse IV, Jan 14 2013
For a spinning black hole the phase transition to positive specific heat happens at a point governed by 2*sqrt(3)-3 (according to a discussion on John Baez's blog), and not at the golden ratio as claimed by Paul Davis. - Peter Luschny, Mar 02 2013
In particular: a black hole with J > (2*sqrt(3)-3) Gm^2/c has positive specific heat, and negative specific heat if J is less, where J is its angular momentum, m is its mass, G is the gravitational constant, and c is the speed of light. For a solar mass black hole, this is 4.08 * 10^41 joule-seconds or a rotation every 1.61 days with the sun's inertia. - Charles R Greathouse IV, Sep 20 2013

			3+2*sqrt(3) = 6.46410161513775458705...

Daniel Starodubtsev, Table of n, a(n) for n = 1..10000
John Baez, Black Holes and The Golden Ratio, Mar 01 2013.
Index entries for algebraic numbers, degree 2.

Cf. A002194 (decimal expansion of sqrt(3)), A010469 (decimal expansion of sqrt(12)), A010696 (repeat 2, 6).

Circs[n_] := With[{r = Sin[Pi/n]/(1 - Sin[Pi/n])}, Graphics[Append[Table[Circle[(r + 1) {Sin[2 Pi k/n], Cos[2 Pi k/n]}, r], {k, n}], {Blue, Circle[{0, 0}, 1]}]]]; Circs[3] (* Charles R Greathouse IV, Jan 14 2013 *)

3+2*sqrt(3) \\ Charles R Greathouse IV, Jan 14 2013

Equals Sum_{n>=1} (sqrt(3)/2)^n = (sqrt(3)/2)/(1 - (sqrt(3)/2)). - Fred Daniel Kline, Mar 03 2014

A194415 Numbers m such that Sum_{k=1..m} (<1/3 + k*r> - ) < 0, where r=sqrt(3) and < > denotes fractional part.

Original entry on oeis.org

1, 2, 4, 5, 8, 16, 17, 19, 20, 23, 31, 32, 34, 35, 38, 46, 47, 49, 50, 53, 56, 57, 58, 59, 60, 61, 62, 64, 65, 68, 71, 72, 73, 74, 75, 76, 77, 79, 80, 83, 86, 87, 88, 89, 90, 91, 92, 94, 95, 98, 101, 102, 103, 104, 105, 106, 107, 109, 110, 112, 113, 114, 115, 116

Offset: 1

Clark Kimberling, Aug 24 2011

nonn

See A194368.

Cf. A002194, A194368, A194416, A194417, A194418.

r = Sqrt[3]; c = 1/3;
x[n_] := Sum[FractionalPart[k*r], {k, 1, n}]
y[n_] := Sum[FractionalPart[c + k*r], {k, 1, n}]
t1 = Table[If[y[n] < x[n], 1, 0], {n, 1, 150}];
Flatten[Position[t1, 1]]           (* A194415 *)
t2 = Table[If[y[n] == x[n], 1, 0], {n, 1, 300}];
Flatten[Position[t2, 1]]           (* A194416 *)
%/3                                (* A194417 *)
t3 = Table[If[y[n] > x[n], 1, 0], {n, 1, 500}];
Flatten[Position[t3, 1]]           (* A194418 *)

cp's OEIS Frontend

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

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI

PARI

PARI

PARI

Formula

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

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Formula

A381671 Decimal expansion of the isoperimetric quotient of a regular tetrahedron.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Mathematica

Formula

A384141 Decimal expansion of the surface area of an elongated pentagonal bipyramid with unit edge.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Formula

A384473 Decimal expansion of the middle interior angle (in degrees) in Albrecht Dürer's approximate construction of the regular pentagon.

Views

Author

Keywords

Examples

References

Links

Crossrefs

Programs

Mathematica

Formula

A384474 Decimal expansion of the middle interior angle (in radians) in Albrecht Dürer's approximate construction of the regular pentagon.

Views

Author

Keywords

Examples

References

Links

Crossrefs

Programs

Mathematica

Formula

A010504 Decimal expansion of square root of 51.

Views

Author

Keywords

Comments

Examples

Links

A165922 Decimal expansion of 2sqrt(3)/(9Pi).