A256780 -id:A256780 - cp's OEIS frontend

A001877 Number of divisors of n of the form 5k+2; a(0) = 0.

0, 0, 1, 0, 1, 0, 1, 1, 1, 0, 1, 0, 2, 0, 2, 0, 1, 1, 1, 0, 1, 1, 2, 0, 2, 0, 1, 1, 2, 0, 1, 0, 2, 0, 2, 1, 2, 1, 1, 0, 1, 0, 3, 0, 2, 0, 1, 1, 2, 1, 1, 1, 2, 0, 2, 0, 2, 1, 1, 0, 2, 0, 2, 1, 2, 0, 2, 1, 2, 0, 2, 0, 3, 0, 2, 0, 1, 2, 1, 0, 1, 1, 2, 0, 4, 1, 1, 1

Offset: 0

N. J. A. Sloane

T. D. Noe, Table of n, a(n) for n = 0..10000
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. A001876, A001878, A001899.

Cf. A001620, A256780.

Join[{0}, Table[d = Divisors[n]; Length[Select[d, Mod[#, 5] == 2 &]], {n, 100}]] (* T. D. Noe, Aug 10 2012 *)
Table[Count[Divisors[n],?(Mod[#,5]==2&)],{n,0,90}] (* _Harvey P. Dale, May 20 2017 *)

a(n) = if (n==0, 0, sumdiv(n, d, (d % 5)==2)); \\ Michel Marcus, Feb 28 2021

G.f.: Sum_{n>=0} x^(5n+2)/(1-x^(5n+2)).
G.f.: Sum_{n>=1} x^(2*n)/(1-x^(5*n)). - Joerg Arndt, Jan 30 2011
Sum_{k=1..n} a(k) = n*log(n)/5 + c*n + O(n^(1/3)*log(n)), where c = gamma(2,5) - (1 - gamma)/5 = A256780 - (1 - A001620)/5 = 0.105832... (Smith and Subbarao, 1981). - Amiram Eldar, Nov 25 2023

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

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.

A218445 a(n) = Sum_{k>=0} floor(n/(5*k + 2)).

Original entry on oeis.org

0, 0, 1, 1, 2, 2, 3, 4, 5, 5, 6, 6, 8, 8, 10, 10, 11, 12, 13, 13, 14, 15, 17, 17, 19, 19, 20, 21, 23, 23, 24, 24, 26, 26, 28, 29, 31, 32, 33, 33, 34, 34, 37, 37, 39, 39, 40, 41, 43, 44, 45, 46, 48, 48, 50, 50, 52, 53, 54, 54, 56, 56, 58, 59, 61, 61, 63, 64, 66, 66, 68, 68, 71, 71, 73, 73, 74, 76, 77, 77, 78

Offset: 0

Benoit Cloitre, Oct 28 2012

Seiichi Manyama, Table of n, a(n) for n = 0..10000
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.

Partial sums of A001877.
Cf. A218444, A218446, A218447.
Cf. A001620, A256780.

Table[Sum[Floor[n/(5k+2)],{k,0,n}],{n,0,80}] (* Harvey P. Dale, Dec 08 2022 *)

A218445[n]:=sum(floor(n/(5*k+2)),k,0,n)$
makelist(A218445[n],n,0,80); /* Martin Ettl, Oct 29 2012 */

PARI
```
a(n)=sum(k=0,n\5,(n\(5*k+2)))
```

a(n) = n*log(n)/5 + c*n + O(n^(1/3)*log(n)), where c = gamma(2,5) - (1 - gamma)/5 = A256780 - (1 - A001620)/5 = 0.105832... (Smith and Subbarao, 1981). - Amiram Eldar, Apr 20 2025

cp's OEIS Frontend

A001877 Number of divisors of n of the form 5k+2; a(0) = 0.

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

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

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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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