A256782 -id:A256782 - 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

A188170 The number of divisors d of n of the form d == 3 (mod 8).

Original entry on oeis.org

0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 1, 1, 0, 0, 1, 0, 0, 1, 1, 0, 1, 1, 0, 1, 0, 0, 2, 0, 0, 1, 0, 0, 2, 0, 1, 1, 0, 1, 1, 0, 0, 1, 1, 1, 1, 0, 0, 1, 0, 0, 2, 0, 0, 2, 1, 0, 2, 0, 1, 1, 0, 0, 1, 0, 0, 2, 1, 0, 1, 1, 0, 1, 0, 0, 2, 1, 1, 1, 0, 0, 2, 0, 1, 1, 0, 1, 1, 1, 0, 1

Offset: 1

R. J. Mathar, Mar 23 2011

a(3n) >= 1 as the divisor d=3 contributes to the count then.

Michael De Vlieger, Table of n, a(n) for n = 1..10000
Michael D. Hirschhorn, The number of representations of a number by various forms, Discrete Mathematics 298 (2005), 205-211.
R. A. Smith and M. V. Subbarao, The average number of divisors in an arithmetic progression, Canadian Mathematical Bulletin, Vol. 24, No. 1 (1981), pp. 37-41.

Cf. A001842, A002325, A188169, A188171, A188172.

Cf. A001620, A256782.

sigmamr := proc(n,m,r) local a,d ; a := 0 ; for d in numtheory[divisors](n) do if modp(d,m) = r then a := a+1 ; end if; end do: a; end proc:
A188170 := proc(n) sigmamr(n,8,3) ; end proc:

Table[Count[Divisors[n],?(Mod[#,8]==3&)],{n,100}] (* _Harvey P. Dale, Jul 08 2013 *)

a(n) = sumdiv(n, d, (d%8) == 3); \\ Michel Marcus, Nov 05 2018

a(n) + A188172(n) = A001842(n).
A188169(n) + a(n) - A188171(n) - A188172(n) = A002325(n).
G.f.: Sum_{k>=1} x^(3*k)/(1 - x^(8*k)). - Ilya Gutkovskiy, Sep 11 2019
Sum_{k=1..n} a(k) = n*log(n)/8 + c*n + O(n^(1/3)*log(n)), where c = gamma(3,8) - (1 - gamma)/8 = A256782 - (1 - A001620)/8 = 0.0314716... (Smith and Subbarao, 1981). - Amiram Eldar, Nov 25 2023

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)

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.

cp's OEIS Frontend

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

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Formula

A188170 The number of divisors d of n of the form d == 3 (mod 8).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Formula

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

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Formula

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

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Formula

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

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Formula

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

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Formula

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

Views

Author

Keywords

Examples

Links

Crossrefs