A256166 -id:A256166 - cp's OEIS frontend

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

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))

A115252 Decimal expansion of -(Pilog((sqrt(2Pi)*Gamma(3/4))/Gamma(1/4)))/2.

Original entry on oeis.org

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

Offset: 0

Eric W. Weisstein, Jan 17 2006

This sequence (its negated version) is also the decimal expansion of the first Malmsten integral int_{x=1..infinity} log(log(x))/(1 + x^2) dx = int_{x=0..1} log(log(1/x))/(1 + x^2) dx = int_{x=0..infinity} 0.5*log(x)/cosh(x) dx = int_{x=Pi/4..Pi/2} log(log(tan(x))) dx = (1/2)*Pi*log(2) + (3/4)*Pi*log(Pi) - Pi*log(Gamma(1/4)). - Iaroslav V. Blagouchine, Mar 29 2015

			0.26044280630098844554009386878972721953181917772314...

G. C. Greubel, Table of n, a(n) for n = 0..5000
I. 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
Eric Weisstein's World of Mathematics, Vardi's Integral

Cf. A256127 (second Malmsten integral), A256128 (third Malmsten integral), A256129 (fourth Malmsten integral), A068466 (Gamma(1/4)), A256166 (log(Gamma(1/4))), A002162 (log 2), A053510 (log Pi).

RealDigits[-Pi/2*Log[Sqrt[2 Pi] Gamma[3/4]/Gamma[1/4]], 10, 111][[1]] (* Robert G. Wilson v, Dec 06 2014 *)

(-Pi*log((sqrt(2*Pi)*gamma(3/4))/gamma(1/4)))/2 \\ Michel Marcus, Dec 06 2014

Equals integral_[0..1] log(1/log(1/x))/(1+x^2) dx. - Jean-François Alcover, Jan 28 2015

cp's OEIS Frontend

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

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Maple

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

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Maple

Mathematica

PARI

A115252 Decimal expansion of -(Pi*log((sqrt(2*Pi)*Gamma(3/4))/Gamma(1/4)))/2.

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Examples

Links

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Programs

Mathematica

PARI

Formula

A115252 Decimal expansion of -(Pilog((sqrt(2Pi)*Gamma(3/4))/Gamma(1/4)))/2.