A256165 -id:A256165 - cp's OEIS frontend

A256614 Decimal expansion of log(Gamma(1/16)).

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

Offset: 1

Iaroslav V. Blagouchine, Apr 04 2015

			2.739631621946203418585729650082232089144944407273537...

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

Cf. decimal expansions of log(Gamma(1/k)): A155968 (k=2), A256165 (k=3), A256166 (k=4), A256167 (k=5), A255888 (k=6), A256609 (k=7), A255306 (k=8), A256610 (k=9), A256612 (k=10), A256611 (k=11), A256066 (k=12), A256615 (k=24), A256616 (k=48).

Maple
```
evalf(log(GAMMA(1/16)),100);
```

RealDigits[Log[Gamma[1/16]],10,100][[1]]

PARI
```
log(gamma(1/16))
```

A256615 Decimal expansion of log(Gamma(1/24)).

Original entry on oeis.org

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

Offset: 1

Iaroslav V. Blagouchine, Apr 04 2015

			3.155402877381144722774664455739805690458351588844024...

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

Cf. decimal expansions of log(Gamma(1/k)): A155968 (k=2), A256165 (k=3), A256166 (k=4), 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), A256616 (k=48).

Maple
```
evalf(log(GAMMA(1/24)),100);
```

RealDigits[Log[Gamma[1/24]],10,100][[1]]

PARI
```
log(gamma(1/24))
```

A256616 Decimal expansion of log(Gamma(1/48)).

Original entry on oeis.org

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

Offset: 1

Iaroslav V. Blagouchine, Apr 04 2015

			3.859529085168528678772669493173125038058701527316499...

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

Cf. decimal expansions of log(Gamma(1/k)): A155968 (k=2), A256165 (k=3), A256166 (k=4), 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).

Maple
```
evalf(log(GAMMA(1/48)),100);
```

RealDigits[Log[Gamma[1/48]],10,100][[1]]

PARI
```
log(gamma(1/48))
```

A257958 Decimal expansion of the Digamma function at 1/Pi, negated.

Original entry on oeis.org

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

Offset: 1

Iaroslav V. Blagouchine, May 14 2015

The reference gives an interesting series representation with rational coefficients for Psi(1/Pi) = -log(Pi) - Pi/2 - 1/2 - 1/8 - 1/72 + 1/64 +7/400 + 7/576 + 643/94080 + 103/30720 + ...

			-3.2902139601732240908430908455400190374021932820070161...

Iaroslav V. Blagouchine, Two series expansions for the logarithm of the gamma function involving Stirling numbers and containing only rational coefficients for certain arguments related to 1/Pi, Mathematics of Computation (AMS), 2015.

Cf. A257955, A257957, A257959, A155968, A256165, A256166, A256167, A255888, A256609, A255306, A256610, A256612, A256611, A256066, A256614, A256615, A256616.

Maple
```
evalf(Psi(1/Pi), 120);
```

RealDigits[PolyGamma[1/Pi], 10, 120][[1]]

PARI
```
default(realprecision, 120); psi(1/Pi)
```

Int_0^infinity x*dx/[(x^2+1)(exp(2x)-1)] = -Pi/2-Psi(1/Pi) = -1.5707...+ 3.2902.. = 1.71941... - R. J. Mathar, Aug 14 2023

A257959 Decimal expansion of the Digamma function at 1/2 + 1/Pi, negated.

Original entry on oeis.org

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

Offset: 0

Iaroslav V. Blagouchine, May 14 2015

The reference gives an interesting series representation with rational coefficients for Psi(1/2 + 1/Pi) = -log(Pi) + 1/4 + 1/16 - 5/576 - 13/512 - 569/25600 -539/36864 - 98671/12042240 - 16231/3932160 - ...

			-0.9236326759613377346000263347486747139894893215261027...

Iaroslav V. Blagouchine, Two series expansions for the logarithm of the gamma function involving Stirling numbers and containing only rational coefficients for certain arguments related to 1/Pi, Mathematics of Computation (AMS), 2015.

Cf. A257955, A257957, A257958, A155968, A256165, A256166, A256167, A255888, A256609, A255306, A256610, A256612, A256611, A256066, A256614, A256615, A256616.

Maple
```
evalf(Psi(1/2+1/Pi), 120);
```

RealDigits[PolyGamma[1/2+1/Pi], 10, 120][[1]]

default(realprecision, 120); psi(1/2+1/Pi)

A257957 Decimal expansion of log(Gamma(1/Pi)).

Original entry on oeis.org

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

Offset: 1

Iaroslav V. Blagouchine, May 14 2015

The reference gives an interesting series representation with rational coefficients for log(Gamma(1/Pi)) = (1-1/Pi)*log(Pi) - 1/Pi + log(2)/2 + (1 + 1/4 + 1/12 + 1/32 + 1/75 + 1/144 + 13/2880 + 157/46080 + ...)/(2*Pi).

The value log(Gamma(1/Pi)) is also intimately related to integral_{x=0..1} arctan(arctanh(x))/x (A257963).

			1.0336461257655827064855374553316178667100308701595988...

Iaroslav V. Blagouchine, Two series expansions for the logarithm of the gamma function involving Stirling numbers and containing only rational coefficients for certain arguments related to 1/Pi, Mathematics of Computation (AMS), 2015.

Cf. A257955, A257963, A257958, A257959, A155968, A256165, A256166, A256167, A255888, A256609, A255306, A256610, A256612, A256611, A256066, A256614, A256615, A256616.

Maple
```
evalf(log(GAMMA(1/Pi)), 120);
```

RealDigits[Log[Gamma[1/Pi]], 10, 120][[1]]

default(realprecision, 120); log(gamma(1/Pi))

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)).

cp's OEIS Frontend

A256614 Decimal expansion of log(Gamma(1/16)).

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A256615 Decimal expansion of log(Gamma(1/24)).

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A256616 Decimal expansion of log(Gamma(1/48)).

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A257958 Decimal expansion of the Digamma function at 1/Pi, negated.

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Maple

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Formula

A257959 Decimal expansion of the Digamma function at 1/2 + 1/Pi, negated.

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Maple

Mathematica

PARI

A257957 Decimal expansion of log(Gamma(1/Pi)).

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Author

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Comments

Examples

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Crossrefs

Programs

Maple

Mathematica

PARI

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

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Programs

Maple