A228725 -id:A228725 - cp's OEIS frontend

A256425 Decimal expansion of the generalized Euler constant gamma(1,3).

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

Offset: 0

N. J. A. Sloane, Apr 09 2015

			0.67780716378423221053372461245491438316931257963255620415268...

Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Section 1.5.3, p. 32.

G. C. Greubel, Table of n, a(n) for n = 0..5000
D. H. Lehmer, Euler constants for arithmetic progressions, Acta Arith. 27 (1975), 125-142. [gamma=A001620, gamma(0,2)=A239097, gamma(1,2)=A228725, N_k=A007947, gamma(1,3)=A256425, the present sequence]

Cf. A001620, A002391, A007947, A047787, A228725, A239097.

RealDigits[EulerGamma/3 + Pi*Sqrt[3]/18 + Log[3]/6, 10, 50][[1]] (* G. C. Greubel, Jun 08 2017 *)

Euler/3 + Pi*sqrt(3)/18 + log(3)/6 \\ Charles R Greathouse IV, Sep 08 2015

-(psi(1/3) + log(3))/3 \\ Charles R Greathouse IV, Feb 03 2025

Equals gamma/3+Pi*sqrt(3)/18+log(3)/6.

Equals -(psi(1/3) + log(3))/3 = (A047787 - A002391)/3. - Amiram Eldar, Jan 07 2024

A256778 Decimal expansion of the generalized Euler constant gamma(1,4).

Original entry on oeis.org

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

Offset: 0

Jean-François Alcover, Apr 10 2015

			0.71028979306409369731376647579508261030406104249693294...

Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Section 1.5.3, p. 32.

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 (EulerGamma), A016627, A020777, A228725 (gamma(1,2)), A256425 (gamma(1,3)), A256779-A256784 (selection of ruler-and-compass constructible gamma(r,k)).

R:=RealField(100); (2*EulerGamma(R) + Pi(R) + 2*Log(2))/8; // G. C. Greubel, Aug 27 2018

RealDigits[EulerGamma/4 + Pi/8 + Log[2]/4, 10, 103] // First

default(realprecision, 100); (2*Euler + Pi + 2*log(2))/8 \\ G. C. Greubel, Aug 27 2018

Equals (2*EulerGamma + Pi + 2*log(2))/8.
Equals Sum_{n>=0} (1/(4n+1) - 1/2*arctanh(2/(4n+3))).
Equals -(psi(1/4) + log(4))/4 = (A020777 - A016627)/4. - Amiram Eldar, Jan 07 2024

A256784 Decimal expansion of the generalized Euler constant gamma(5,12) (negated).

Original entry on oeis.org

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

Offset: 0

Jean-François Alcover, Apr 10 2015

			-0.0033729493224032970250324948185921946160340346994983953873167...

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 (EulerGamma), A016635, A228725 (gamma(1,2)), A256425 (gamma(1,3)), A256778-A256784 (selection of ruler-and-compass constructible gamma(r,k)).

SetDefaultRealField(RealField(100)); R:= RealField(); EulerGamma(R)/12 + 1/24*(Pi(R)*(2-Sqrt(3)) + 2*(Sqrt(3)+1)*Log(2) + Log(3) - 4*Sqrt(3)*Log(Sqrt(3)+1)); // G. C. Greubel, Aug 27 2018

Join[{0, 0}, RealDigits[-Log[12]/12 - PolyGamma[5/12]/12, 10, 105] // First]

default(realprecision, 100); Euler/12 + 1/24*(Pi*(2-sqrt(3)) + 2*(sqrt(3)+1)*log(2) + log(3) - 4*sqrt(3)*log(sqrt(3)+1)) \\ G. C. Greubel, Aug 27 2018

Equals EulerGamma/12 + 1/24*(Pi*(2-sqrt(3)) + 2*(sqrt(3)+1)*log(2) + log(3) - 4*sqrt(3) * log(sqrt(3)+1)).

Equals -(psi(5/12) + log(12))/12. - Amiram Eldar, Jan 07 2024

A239097 Decimal expansion of -(gamma-log(2))/2.

Original entry on oeis.org

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

Offset: 0

N. J. A. Sloane, Mar 23 2014

Decimal expansion of the generalized Euler constant -gamma(0,2).

			.057965757829206224405360015687887068516670399210165827657456...

G. C. Greubel, Table of n, a(n) for n = 0..10000
D. H. Lehmer, Euler constants for arithmetical progressions, Collection of articles in memory of Juriĭ Vladimirovič Linnik. Acta Arith. 27 (1975), 125--142. MR0369233 (51 #5468). See p. 128.

Cf. A001620, A228725.

SetDefaultRealField(RealField(100)); R:= RealField();
(Log(2) - EulerGamma(R))/2; // G. C. Greubel, Aug 28 2018

Join[{0}, RealDigits[(Log[2] - EulerGamma)/2, 10, 100][[1]]] (* G. C. Greubel, Aug 28 2018 *)

(log(2)-Euler)/2 \\ Charles R Greathouse IV, Mar 25 2014

From Amiram Eldar, Jun 30 2020: (Start)
Equals Sum_{k>=1} zeta(2*k+1)/((2*k+1)*2^(2*k+1)).
Equals Sum_{k>=1} arctanh(1/(2*k)) - 1/(2*k). (End)

A256779 Decimal expansion of the generalized Euler constant gamma(1,5).

Original entry on oeis.org

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

Offset: 0

Jean-François Alcover, Apr 10 2015

			0.735920396831617584189289725844752893059997383987625...

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 (EulerGamma), A016628, A200135, A228725 (gamma(1,2)), A256425 (gamma(1,3)), A256778-A256784 (selection of ruler-and-compass constructible gamma(r,k)).

SetDefaultRealField(RealField(100)); R:= RealField(); EulerGamma(R)/5 + Pi(R)/10*Sqrt(1 + 2/Sqrt(5)) + Log(5)/20 + Sqrt(5)/10*Log((1 + Sqrt(5))/2); // G. C. Greubel, Aug 28 2018

RealDigits[-Log[5]/5 - PolyGamma[1/5]/5, 10, 105] // First

Euler/5 + Pi/10*sqrt(1 + 2/sqrt(5)) + log(5)/20 + sqrt(5)/10*log((1 + sqrt(5))/2) \\ Michel Marcus, Apr 10 2015

Equals EulerGamma/5 + Pi/10*sqrt(1 + 2/sqrt(5)) + log(5)/20 + sqrt(5)/10*log((1 + sqrt(5))/2).
Equals Sum_{n>=0} (1/(5n+1) - 2/5*arctanh(5/(10n+7))).
Equals -(psi(1/5) + log(5))/5 = (A200135 - A016628)/5. - Amiram Eldar, Jan 07 2024

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

Original entry on oeis.org

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

A256780 Decimal expansion of the generalized Euler constant gamma(2,5).

Original entry on oeis.org

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

Offset: 0

Jean-François Alcover, Apr 10 2015

			0.190389326430203154225983229764268163260151948448458487...

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 (EulerGamma), A016628, A200136, A228725 (gamma(1,2)), A256425 (gamma(1,3)), A256778-A256784 (selection of ruler-and-compass constructible gamma(r,k)).

SetDefaultRealField(RealField(100)); R:= RealField(); EulerGamma(R)/5 + Pi(R)/10*Sqrt(1 - 2/Sqrt(5)) + Log(5)/20 - Sqrt(5)/10*Log((1 + Sqrt(5))/2); // G. C. Greubel, Aug 28 2018

RealDigits[-Log[5]/5 - PolyGamma[2/5]/5, 10, 105] // First

Euler/5 + Pi/10*sqrt(1 - 2/sqrt(5)) + log(5)/20 - sqrt(5)/10*log((1 + sqrt(5))/2) \\ Michel Marcus, Apr 10 2015

Equals EulerGamma/5 + Pi/10*sqrt(1 - 2/sqrt(5)) + log(5)/20 - sqrt(5)/10*log((1 + sqrt(5))/2).
Equals Sum_{n>=0} (1/(5n+2) - 2/5*arctanh(5/(10n+9))).
Equals -(psi(2/5) + log(5))/5 = (A200136 - A016628)/5. - Amiram Eldar, Jan 07 2024

A256781 Decimal expansion of the generalized Euler constant gamma(1,8).

Original entry on oeis.org

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

Offset: 0

Jean-François Alcover, Apr 10 2015

			0.788631390202002367443880819838976661978118204921...

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 (EulerGamma), A016631, A228725 (gamma(1,2)), A250129, A256425 (gamma(1,3)), A256778-A256784 (selection of ruler-and-compass constructible gamma(r,k)).

SetDefaultRealField(RealField(100)); R:= RealField(); EulerGamma(R)/8 + (1/8)*(Pi(R)/2*(Sqrt(2)+1) + Log(2) + Sqrt(2)*Log(Sqrt(2) + 1)); // G. C. Greubel, Aug 28 2018

RealDigits[-3/8*Log[2] - PolyGamma[1/8]/8, 10, 105] // First

Euler/8 + 1/8*(Pi/2*(sqrt(2)+1) + log(2) + sqrt(2)*log(sqrt(2) + 1)) \\ Michel Marcus, Apr 10 2015

Equals EulerGamma/8 + 1/8*(Pi/2*(sqrt(2)+1) + log(2) + sqrt(2)*log(sqrt(2) + 1)).
Equals Sum_{n>=0} (1/(8n+1) - 1/4*arctanh(4/(8n+5))).
Equals -(psi(1/8) + log(8))/8 = -(A250129 + A016631)/8. - Amiram Eldar, Jan 07 2024

A256782 Decimal expansion of the generalized Euler constant gamma(3,8).

Original entry on oeis.org

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

Offset: 0

Jean-François Alcover, Apr 10 2015

			0.08431968843316295593904035680375480012812437382591706852303...

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 (EulerGamma), A016631, A228725 (gamma(1,2)), A256425 (gamma(1,3)), A256778-A256784 (selection of ruler-and-compass constructible gamma(r,k)), A354633.

SetDefaultRealField(RealField(100)); R:= RealField(); EulerGamma(R)/8 + (1/8)*(Pi(R)/2*(Sqrt(2)-1) + Log(2) - Sqrt(2)*Log(Sqrt(2)+1)); // G. C. Greubel, Aug 28 2018

Join[{0}, RealDigits[-3/8*Log[2] - PolyGamma[3/8]/8, 10, 104] // First]

default(realprecision, 100); Euler/8 + 1/8*(Pi/2*(sqrt(2)-1) + log(2) - sqrt(2)*log(sqrt(2)+1)) \\ G. C. Greubel, Aug 28 2018

Equals EulerGamma/8 + 1/8*(Pi/2*(sqrt(2)-1) + log(2) - sqrt(2)*log(sqrt(2)+1)).

Equals -(psi(3/8) + log(8))/8 = -(A354633 + A016631)/8. - Amiram Eldar, Jan 07 2024

A256845 Decimal expansion of the generalized Euler constant gamma(2,4).

Original entry on oeis.org

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

Offset: 0

Jean-François Alcover, Apr 11 2015

			0.1443039162253832151516280225206006077605398339849808997...

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).

R:= RealField(100); EulerGamma(R)/4; // G. C. Greubel, Aug 28 2018

RealDigits[EulerGamma/4, 10, 107] // First

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

-log(4)/4 - PolyGamma(1/2)/4 = EulerGamma/4
From Amiram Eldar, Jul 21 2020: (Start)
Equals -Integral_{x=0..oo} e^(-x^2)*x*log(x) dx.
Equals Integral_{x=0..oo} (e^(-x^4) - e^(-x^2))/x dx. (End)

cp's OEIS Frontend

A256425 Decimal expansion of the generalized Euler constant gamma(1,3).

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A256778 Decimal expansion of the generalized Euler constant gamma(1,4).

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A256784 Decimal expansion of the generalized Euler constant gamma(5,12) (negated).

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A239097 Decimal expansion of -(gamma-log(2))/2.

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A256779 Decimal expansion of the generalized Euler constant gamma(1,5).

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

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A256780 Decimal expansion of the generalized Euler constant gamma(2,5).

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