A256846 -id:A256846 - cp's OEIS frontend

A001842 Expansion of Sum_{n>=0} x^(4n+3)/(1 - x^(4n+3)).

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

Offset: 0

N. J. A. Sloane

Number of divisors of n of the form 4*k+3. - Reinhard Zumkeller, Apr 18 2006

G. H. Hardy, Ramanujan: twelve lectures on subjects suggested by his life and work, Cambridge, University Press, 1940, p. 132.
James J. Tattersall, Elementary Number Theory in Nine Chapters, Cambridge University Press, 1999, page 244.

T. D. Noe, Table of n, a(n) for n = 0..10000
Michael Gilleland, Some Self-Similar Integer Sequences.
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. A001227, A001620, A001826, A072437, A256846.

with(numtheory): seq(add(binomial(d,3) mod 2, d in divisors(n)), n=0..100); # Ridouane Oudra, Nov 19 2019

Join[{0}, Table[d = Divisors[n]; Length[Select[d, Mod[#, 4] == 3 &]], {n, 100}]] (* T. D. Noe, Aug 10 2012 *)
a[n_] := DivisorSum[n, 1 &, Mod[#, 4] == 3 &]; a[0] = 0; Array[a, 100, 0] (* Amiram Eldar, Nov 25 2023 *)

a(n) = if(n<1, 0, sumdiv(n, d, d%4 == 3)); \\ Amiram Eldar, Nov 25 2023

a(A072437(n)) = 0. - Benoit Cloitre, Apr 24 2003
a(n) = A001227(n) - A001826(n). - Reinhard Zumkeller, Apr 18 2006
G.f.: Sum_{k>=1} x^(3*k)/(1 - x^(4*k)). - Ilya Gutkovskiy, Sep 11 2019
a(n) = Sum_{d|n} (binomial(d,3) mod 2). - Ridouane Oudra, Nov 19 2019
Sum_{k=1..n} a(k) = n*log(n)/4 + c*n + O(n^(1/3)*log(n)), where c = gamma(3,4) - (1 - gamma)/4 = A256846 - (1 - A001620)/4 = -0.180804... (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)

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.

A293513 Number of proper divisors of n of the form 4k+3.

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, Oct 19 2017

Antti Karttunen, Table of n, a(n) for n = 1..16384
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, A091954, A121262, A293451, A293903.

a[n_] := DivisorSum[n, 1 &, # < n && Mod[#, 4] == 3 &]; Array[a, 100] (* Amiram Eldar, Nov 25 2023 *)

PARI
```
A293513(n) = sumdiv(n,d,(d
				
```

a(n) = Sum_{d|n, d

a(n) = A091954(n) - A293451(n).

a(n) = A001842(n) - A121262(n+1).

G.f.: Sum_{k>=1} x^(8*k-2) / (1 - x^(4*k-1)). - Ilya Gutkovskiy, Apr 14 2021

Sum_{k=1..n} a(k) = n*log(n)/4 + c*n + O(n^(1/3)*log(n)), where c = gamma(3,4) - (2 - gamma)/4 = A256846 - (2 - A001620)/4 = -0.430804... (Smith and Subbarao, 1981). - Amiram Eldar, Nov 25 2023

Showing 1-9 of 9 results.

cp's OEIS Frontend

A001842 Expansion of Sum_{n>=0} x^(4*n+3)/(1 - x^(4*n+3)).

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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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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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Magma

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

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A001842 Expansion of Sum_{n>=0} x^(4n+3)/(1 - x^(4n+3)).