A256165 -id:A256165 - cp's OEIS frontend

A073005 Decimal expansion of Gamma(1/3).

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

Offset: 1

Robert G. Wilson v, Aug 03 2002

Nesterenko proves that this constant is transcendental (he cites Chudnovsky as the first to show this); in fact it is algebraically independent of Pi and exp(sqrt(3)*Pi) over Q. - Charles R Greathouse IV, Nov 11 2013

			Gamma(1/3) = 2.6789385347077476336556929409746776441286893779573011009...

H. B. Dwight, Tables of Integrals and other Mathematical Data. 860.18, 860.19 in Definite Integrals. New York, U.S.A.: Macmillan Publishing, 1961, p. 230.
Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Section 1.5.4, p. 33.
Jerome Spanier and Keith B. Oldham, "Atlas of Functions", Hemisphere Publishing Corp., 1987, chapter 43, equation 43:4:8 at page 413.

Harry J. Smith, Table of n, a(n) for n = 1..1000
Alessandro Languasco and Pieter Moree, Euler constants from primes in arithmetic progression, arXiv:2406.16547 [math.NT], 2024. See p. 8.
Yu. V. Nesterenko, Modular functions and transcendence questions, Sbornik: Mathematics 187:9 (1996), pp. 1319-1348. (English translation)
Andrea Pinos, Gamma of reciprocal by Laplace.
Simon Plouffe, GAMMA(1/3).
Wikipedia, Particular values of the Gamma function: General rational arguments.
Index entries for transcendental numbers.
Index to sequences related to the Gamma function.

Cf. A073006, A202623, A203145, A256165.

R:= RealField(100); SetDefaultRealField(R); Gamma(1/3); // G. C. Greubel, Mar 10 2018

Mathematica
```
RealDigits[ N[ Gamma[1/3], 110]][[1]]
```

default(realprecision, 1080); x=gamma(1/3); for (n=1, 1000, d=floor(x); x=(x-d)*10; write("b073005.txt", n, " ", d)); \\ Harry J. Smith, Apr 19 2009

this * A073006 = A186706. - R. J. Mathar, Jan 15 2021
From Amiram Eldar, Jun 25 2021: (Start)
Equals 2^(7/9) * Pi^(1/3) * K((sqrt(3)-1)/(2*sqrt(2)))^(1/3)/3^(1/12), where K is the complete elliptic integral of the first kind.
Equals 2^(7/9) * Pi^(2/3) /(AGM(2, sqrt(2+sqrt(3)))^(1/3) * 3^(1/12)), where AGM is the arithmetic-geometric mean. (End)
From Andrea Pinos, Aug 12 2023: (Start)
Equals Integral_{x=0..oo} 3*exp(-(x^3)) dx = 3*A202623.
General result: Gamma(1/n) = Integral_{x=0..oo} n*exp(-(x^n)) dx. (End)
Equals 3*A202623 = exp(A256165). - Hugo Pfoertner, Jun 28 2024
Equals (2^(1/3)*Pi*C*3^(1/2))^(1/3), where C = A118292 = Integral {0..1} 2/sqrt(1-x^3) is the transcendental butterfly constant. - Jan Lügering, Feb 08 2025

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

Original entry on oeis.org

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

A255306 Decimal expansion of log(Gamma(1/8)).

Original entry on oeis.org

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

Offset: 1

Iaroslav V. Blagouchine, Mar 18 2015

			2.0194183575537963453202905211670995899482809521344496...

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

Cf. A203142 (Gamma(1/8)), A255188 (first generalized Stieltjes constant at 1/8, gamma_1(1/8)).

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), A256610 (k=9), A256612 (k=10), A256611 (k=11), A256066 (k=12), A256614 (k=16), A256615 (k=24), A256616 (k=48).

Maple
```
evalf(log(GAMMA(1/8)),100);
```
Mathematica
```
RealDigits[Log[Gamma[1/8]],10,100][[1]]
```
PARI
```
log(gamma(1/8))
```

A255888 Decimal expansion of log(Gamma(1/6)).

Original entry on oeis.org

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

Offset: 1

Iaroslav V. Blagouchine, Mar 18 2015

			1.71673343507824046052784630958793075727937748710540556...

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

Cf. A175379 (Gamma(1/6)), A254349 (first generalized Stieltjes constant at 1/6, gamma_1(1/6)).

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), A256616 (k=48).

evalf(log(GAMMA(1/6)),100);
evalf((1/2)*log(3)-(1/3)*log(2)-(1/2)*log(Pi)+2*log(GAMMA(1/3)), 120);

Mathematica
```
RealDigits[Log[Gamma[1/6]],10,100][[1]]
```
PARI
```
log(gamma(1/6))
```

Equals (1/2)*log(3) - (1/3)*log(2) - (1/2)*log(Pi) + 2*log(Gamma(1/3)).

A256066 Decimal expansion of log(Gamma(1/12)).

Original entry on oeis.org

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

Offset: 1

Iaroslav V. Blagouchine, Mar 18 2015

			2.44229731118288975091554935219408858208684110709150...

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

Cf. A203140 (Gamma(1/12)), A256165 (log(Gamma(1/3))), A256166 (log(Gamma(1/4))), A256167 (log(Gamma(1/5))), A255888 (log(Gamma(1/6))), A255306 (log(Gamma(1/8))), A255189 (first generalized Stieltjes constant at 1/12, gamma_1(1/12)).

evalf(log(GAMMA(1/12)),100);
evalf(-(1/4)*log(2)+(3/8)*log(3)+(1/2)*log(1+sqrt(3))-(1/2)*log(Pi)+log(GAMMA(1/4))+log(GAMMA(1/3)), 100);

RealDigits[Log[Gamma[1/12]],10,100][[1]]

PARI
```
log(gamma(1/12))
```

Equals -(1/4)*log(2) + (3/8)*log(3) + (1/2)*log(1+sqrt(3)) - (1/2)*log(Pi) + log(Gamma(1/4)) + log(Gamma(1/3)).

A256167 Decimal expansion of log(Gamma(1/5)).

Original entry on oeis.org

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

Offset: 1

Iaroslav V. Blagouchine, Mar 17 2015

			1.524063822430784524881056493926302192565933737406403...

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

Cf. A175380 (Gamma(1/5)), A251866 (gamma_1(1/5)).

Cf. decimal expansions of log(Gamma(1/k)): A155968 (k=2), A256165 (k=3), A256166 (k=4), 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(1/5)),120); # Vaclav Kotesovec, Mar 17 2015

RealDigits[Log[Gamma[1/5]],10,105][[1]] (* Vaclav Kotesovec, Mar 17 2015 *)

log(gamma(1/5)) \\ Michel Marcus, Mar 17 2015

A256609 Decimal expansion of log(Gamma(1/7)).

Original entry on oeis.org

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

Offset: 1

Iaroslav V. Blagouchine, Apr 04 2015

			1.879169271595835836455956409345071054399542621720334...

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), A255306 (k=8), A256610 (k=9), A256612 (k=10), A256611 (k=11), A256066 (k=12), A256614 (k=16), A256615 (k=24), A256616 (k=48).

Maple
```
evalf(log(GAMMA(1/7)),100);
```
Mathematica
```
RealDigits[Log[Gamma[1/7]],10,100][[1]]
```
PARI
```
log(gamma(1/7))
```

A256610 Decimal expansion of log(Gamma(1/9)).

Original entry on oeis.org

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

Offset: 1

Iaroslav V. Blagouchine, Apr 04 2015

			2.142731800376693104880407884896452638386509294121518...

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), A256612 (k=10), A256611 (k=11), A256066 (k=12), A256614 (k=16), A256615 (k=24), A256616 (k=48).

Maple
```
evalf(log(GAMMA(1/9)),100);
```
Mathematica
```
RealDigits[Log[Gamma[1/9]],10,100][[1]]
```
PARI
```
log(gamma(1/9))
```

A256611 Decimal expansion of log(Gamma(1/11)).

Original entry on oeis.org

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

Offset: 1

Iaroslav V. Blagouchine, Apr 04 2015

			2.351934610798792760463680957649386963365693018602581...

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), A256066 (k=12), A256614 (k=16), A256615 (k=24), A256616 (k=48).

Maple
```
evalf(log(GAMMA(1/11)),100);
```

RealDigits[Log[Gamma[1/11]],10,100][[1]]

PARI
```
log(gamma(1/11))
```

A256612 Decimal expansion of log(Gamma(1/10)).

Original entry on oeis.org

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

Offset: 1

Iaroslav V. Blagouchine, Apr 04 2015

			2.252712651734205959869701646368495118615627222294953...

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), A256611 (k=11), A256066 (k=12), A256614 (k=16), A256615 (k=24), A256616 (k=48).

Maple
```
evalf(log(GAMMA(1/10)),100);
```

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

PARI
```
log(gamma(1/10))
```

cp's OEIS Frontend

A073005 Decimal expansion of Gamma(1/3).

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

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A255306 Decimal expansion of log(Gamma(1/8)).

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A255888 Decimal expansion of log(Gamma(1/6)).

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A256066 Decimal expansion of log(Gamma(1/12)).

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A256167 Decimal expansion of log(Gamma(1/5)).

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Maple

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A256609 Decimal expansion of log(Gamma(1/7)).

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