A256845 -id:A256845 - 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

A256846 Decimal expansion of the generalized Euler constant gamma(3,4) (negated).

Original entry on oeis.org

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

Offset: 0

Jean-François Alcover, Apr 11 2015

			-0.07510837033335461230189437002479311074523130734684351439...

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),
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)/4 - Pi(R)/8 - Log(4)/4 + Log(8)/4; // G. C. Greubel, Aug 28 2018

Join[{0}, RealDigits[-Log[4]/4 - PolyGamma[3/4]/4, 10, 104] // First ]

default(realprecision, 100); Euler/4 - Pi/8 - log(4)/4 + log(8)/4 \\ G. C. Greubel, Aug 28 2018

-log(4)/4 - PolyGamma(3/4)/4 = EulerGamma/4 - Pi/8 - log(4)/4 + log(8)/4

A256848 Decimal expansion of the generalized Euler constant gamma(3,5) (negated).

Original entry on oeis.org

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

Offset: 0

Jean-François Alcover, Apr 11 2015

			-0.013763739708181991968019076883991139603013419915782102729...

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),
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)/5 + Pi(R)/(10*Sqrt(2*(5+Sqrt(5)))) - Pi(R)/(2*Sqrt(10*(5+Sqrt(5)))) + Log(5)/20 + Log((5-Sqrt(5))/(5+Sqrt(5)))/(4*Sqrt(5)); // G. C. Greubel, Aug 28 2018

Join[{0}, RealDigits[-Log[5]/5 - PolyGamma[3/5]/5, 10, 108] // First  ]

default(realprecision, 100); Euler/5 + Pi/(10*sqrt(2*(5+sqrt(5)))) - Pi/(2*sqrt(10*(5+sqrt(5)))) + log(5)/20 + log((5-sqrt(5))/(5+sqrt(5)))/(4*sqrt(5)) \\ G. C. Greubel, Aug 28 2018

Equals -log(5)/5 - PolyGamma(3/5)/5.

Equals EulerGamma/5 + Pi/(10*sqrt(2*(5+sqrt(5)))) - Pi/(2*sqrt(10*(5+sqrt(5)))) + log(5)/20 + log((5-sqrt(5))/(5+sqrt(5)))/(4*sqrt(5)).

A256849 Decimal expansion of the generalized Euler constant gamma(4,5) (negated).

Original entry on oeis.org

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

Offset: 0

Jean-François Alcover, Apr 11 2015

			-0.12888586914559238304189234001387044397828817291465897856 ...

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),
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)/5 - Pi(R)/(10*Sqrt(2*(5-Sqrt(5)))) - Pi(R)/(2*Sqrt(10*(5-Sqrt(5)))) + Log(5)/20 - Log(5-Sqrt(5))/(4*Sqrt(5)) + Log(5+Sqrt(5))/( 4*Sqrt(5)); // G. C. Greubel, Aug 28 2018

RealDigits[-Log[5]/5 - PolyGamma[4/5]/5, 10, 107] // First

default(realprecision, 100); Euler/5 - Pi/(10*sqrt(2*(5-sqrt(5)))) - Pi/(2*sqrt(10*(5-sqrt(5)))) + log(5)/20 - log(5-sqrt(5))/(4*sqrt(5)) + log(5+sqrt(5))/(4*sqrt(5)) \\ G. C. Greubel, Aug 28 2018

Equals -log(5)/5 - PolyGamma(4/5)/5.

Equals EulerGamma/5 - Pi/(10*sqrt(2*(5-sqrt(5)))) - Pi/(2*sqrt(10*(5-sqrt(5)))) + log(5)/20 - log(5-sqrt(5))/(4*sqrt(5)) + log(5+sqrt(5))/(4*sqrt(5)).

A256844 Decimal expansion of the generalized Euler constant gamma(3,3) (negated).

Original entry on oeis.org

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

Offset: 0

Jean-François Alcover, Apr 11 2015

			-0.1737988745888589435962443822800410912017770739609419509763...

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),
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 - Log(3)/3; // G. C. Greubel, Aug 28 2018

RealDigits[EulerGamma/3 - Log[3]/3, 10, 105] // First

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

Equals EulerGamma/3 - log(3)/3.

A256847 Decimal expansion of the generalized Euler constant gamma(4,4) (negated).

Original entry on oeis.org

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

Offset: 0

Jean-François Alcover, Apr 11 2015

			-0.202269674054589439556988038208487676277210233195146727358898...

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),
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) - Log(4))/4; // G. C. Greubel, Aug 28 2018

RealDigits[EulerGamma/4 - Log[4]/4, 10, 104] // First

default(realprecision, 100); (Euler - log(4))/4 \\ G. C. Greubel, Aug 28 2018

Equals (EulerGamma - log(4))/4.

A256850 Decimal expansion of the generalized Euler constant gamma(5,5) (negated).

Original entry on oeis.org

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

Offset: 0

Jean-François Alcover, Apr 11 2015

			-0.20644444950651350279884944862875704169668840366571882462...

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),
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) - Log(5))/5; // G. C. Greubel, Aug 28 2018

RealDigits[EulerGamma/5 - Log[5]/5, 10, 102] // First

default(realprecision, 100); (Euler - log(5))/5 \\ G. C. Greubel, Aug 28 2018

Equals (EulerGamma - log(5))/5.

A374774 Decimal expansion of (2 + gamma - log(Pi))/4.

Original entry on oeis.org

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

Offset: 0

Stefano Spezia, Jul 19 2024

			0.3581214447630331716157711846823359298487161...

Stephane Louboutin, Majorations explicites de |L(1, χ)| (Suite), C. R. Acad. Sci. Paris. 323, pp. 443-446 (1996). (In French). See Theorem 1 at p. 444.

Cf. A374773, A374775.

Cf. A000796, A001620, A053510, A256845.

RealDigits[(2+EulerGamma-Log[Pi])/4,10,100][[1]]

Equals A374775/2.

A374776 Decimal expansion of (2 + gamma - log(Pi/4))/4.

Original entry on oeis.org

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

Offset: 0

Stefano Spezia, Jul 19 2024

			0.7046950350430058263243872454114242138864661979...

Stephane Louboutin, Majorations explicites de |L(1, χ)| (Suite), C. R. Acad. Sci. Paris. 323, pp. 443-446 (1996). (In French). See Theorem 2 at p. 444.

Cf. A374775.

Cf. A000796, A001620, A094640, A256845.

RealDigits[(2+EulerGamma-Log[Pi/4])/4,10,100][[1]]

cp's OEIS Frontend

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

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A256846 Decimal expansion of the generalized Euler constant gamma(3,4) (negated).

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A256848 Decimal expansion of the generalized Euler constant gamma(3,5) (negated).

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A256849 Decimal expansion of the generalized Euler constant gamma(4,5) (negated).

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A256844 Decimal expansion of the generalized Euler constant gamma(3,3) (negated).

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A256847 Decimal expansion of the generalized Euler constant gamma(4,4) (negated).

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A256850 Decimal expansion of the generalized Euler constant gamma(5,5) (negated).

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