cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-10 of 11 results. Next

A001822 Expansion of Sum_{n>=0} x^(3n+2)/(1-x^(3n+2)).

Original entry on oeis.org

0, 1, 0, 1, 1, 1, 0, 2, 0, 2, 1, 1, 0, 2, 1, 2, 1, 1, 0, 3, 0, 2, 1, 2, 1, 2, 0, 2, 1, 2, 0, 3, 1, 2, 2, 1, 0, 2, 0, 4, 1, 2, 0, 3, 1, 2, 1, 2, 0, 3, 1, 2, 1, 1, 2, 4, 0, 2, 1, 3, 0, 2, 0, 3, 2, 2, 0, 3, 1, 4, 1, 2, 0, 2, 1, 2, 2, 2, 0, 5, 0, 2, 1, 2, 2, 2, 1, 4, 1, 2, 0, 3, 0, 2, 2, 3, 0, 3, 1, 4, 1, 2, 0, 4, 2
Offset: 1

Views

Author

Keywords

Comments

a(n) is the number of positive divisors of n of the form 3k+2. If r(n) denotes the number of representations of n by the quadratic form j^2+i*j+i^2, then r(n)= 6 *(A001817(n)-a(n)). - Benoit Cloitre, Jun 24 2002

References

  • Bruce C. Berndt,"On a certain theta-function in a letter of Ramanujan from Fitzroy House", Ganita 43 (1992),33-43.

Crossrefs

Programs

  • Haskell
    a001822 n = length [d | d <- [2,5..n], mod n d == 0]
    -- Reinhard Zumkeller, Nov 26 2011
  • Maple
    A001822 := proc(n)
        local a,d ;
        a := 0 ;
        for d in numtheory[divisors](n) do
            if modp(d,3) = 2 then
                a := a+1 ;
            end if ;
        end do:
        a ;
    end proc:
    seq(A001822(n),n=1..100) ; # R. J. Mathar, Sep 25 2017
  • Mathematica
    a[n_] := DivisorSum[n, Boole[Mod[#, 3] == 2]&]; Array[a, 100] (* Jean-François Alcover, Dec 01 2015 *)
  • PARI
    a(n)=if(n<1, 0, sumdiv(n,d, d%3==2))
    

Formula

Moebius transform is period 3 sequence [0, 1, 0, ...]. - Michael Somos, Sep 20 2005
G.f.: Sum_{k>0} x^(3k-1)/(1-x^(3k-1)) = Sum_{k>0} x^(2k)/(1-x^(3k)). - Michael Somos, Sep 20 2005
a(n) = (A035191(n) - A002324(n)) / 2. - Reinhard Zumkeller, Nov 26 2011
a(n) + A001817(n) + A000005(n/3) = A000005(n), where A000005(.)=0 if the argument is not an integer. - R. J. Mathar, Sep 25 2017
Sum_{k=1..n} a(k) = n*log(n)/3 + c*n + O(n^(1/3)*log(n)), where c = gamma(2,3) - (1 - gamma)/3 = A256843 - (1 - A001620)/3 = -0.0677207... (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

Views

Author

Jean-François Alcover, Apr 11 2015

Keywords

Examples

			0.1443039162253832151516280225206006077605398339849808997...
		

Crossrefs

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).

Programs

  • Magma
    R:= RealField(100); EulerGamma(R)/4; // G. C. Greubel, Aug 28 2018
  • Mathematica
    RealDigits[EulerGamma/4, 10, 107] // First
  • PARI
    default(realprecision, 100); Euler/4 \\ G. C. Greubel, Aug 28 2018
    

Formula

-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

Views

Author

Jean-François Alcover, Apr 11 2015

Keywords

Examples

			-0.07510837033335461230189437002479311074523130734684351439...
		

Crossrefs

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).

Programs

  • Magma
    SetDefaultRealField(RealField(100)); R:= RealField(); EulerGamma(R)/4 - Pi(R)/8 - Log(4)/4 + Log(8)/4; // G. C. Greubel, Aug 28 2018
  • Mathematica
    Join[{0}, RealDigits[-Log[4]/4 - PolyGamma[3/4]/4, 10, 104] // First ]
  • PARI
    default(realprecision, 100); Euler/4 - Pi/8 - log(4)/4 + log(8)/4 \\ G. C. Greubel, Aug 28 2018
    

Formula

-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

Views

Author

Jean-François Alcover, Apr 11 2015

Keywords

Examples

			-0.013763739708181991968019076883991139603013419915782102729...
		

Crossrefs

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).

Programs

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

Formula

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

Views

Author

Jean-François Alcover, Apr 11 2015

Keywords

Examples

			-0.12888586914559238304189234001387044397828817291465897856 ...
		

Crossrefs

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).

Programs

  • Magma
    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
  • Mathematica
    RealDigits[-Log[5]/5 - PolyGamma[4/5]/5, 10, 107] // First
  • PARI
    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
    

Formula

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

Views

Author

Jean-François Alcover, Apr 11 2015

Keywords

Examples

			-0.1737988745888589435962443822800410912017770739609419509763...
		

Crossrefs

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).

Programs

  • Magma
    SetDefaultRealField(RealField(100)); R:= RealField(); EulerGamma(R)/3 - Log(3)/3; // G. C. Greubel, Aug 28 2018
  • Mathematica
    RealDigits[EulerGamma/3 - Log[3]/3, 10, 105] // First
  • PARI
    default(realprecision, 100); Euler/3 - log(3)/3 \\ G. C. Greubel, Aug 28 2018
    

Formula

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

Views

Author

Jean-François Alcover, Apr 11 2015

Keywords

Examples

			-0.202269674054589439556988038208487676277210233195146727358898...
		

Crossrefs

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).

Programs

  • Magma
    SetDefaultRealField(RealField(100)); R:= RealField(); (EulerGamma(R) - Log(4))/4; // G. C. Greubel, Aug 28 2018
  • Mathematica
    RealDigits[EulerGamma/4 - Log[4]/4, 10, 104] // First
  • PARI
    default(realprecision, 100); (Euler - log(4))/4 \\ G. C. Greubel, Aug 28 2018
    

Formula

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

Views

Author

Jean-François Alcover, Apr 11 2015

Keywords

Examples

			-0.20644444950651350279884944862875704169668840366571882462...
		

Crossrefs

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).

Programs

  • Magma
    SetDefaultRealField(RealField(100)); R:= RealField(); (EulerGamma(R) - Log(5))/5; // G. C. Greubel, Aug 28 2018
  • Mathematica
    RealDigits[EulerGamma/5 - Log[5]/5, 10, 102] // First
  • PARI
    default(realprecision, 100); (Euler - log(5))/5 \\ G. C. Greubel, Aug 28 2018
    

Formula

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

A293896 Number of proper divisors of n of the form 3k+2.

Original entry on oeis.org

0, 0, 0, 1, 0, 1, 0, 1, 0, 2, 0, 1, 0, 1, 1, 2, 0, 1, 0, 2, 0, 2, 0, 2, 1, 1, 0, 2, 0, 2, 0, 2, 1, 2, 1, 1, 0, 1, 0, 4, 0, 2, 0, 2, 1, 2, 0, 2, 0, 2, 1, 2, 0, 1, 2, 3, 0, 2, 0, 3, 0, 1, 0, 3, 1, 2, 0, 2, 1, 4, 0, 2, 0, 1, 1, 2, 1, 2, 0, 4, 0, 2, 0, 2, 2, 1, 1, 4, 0, 2, 0, 2, 0, 2, 1, 3, 0, 2, 1, 4, 0, 2, 0, 3, 2
Offset: 1

Views

Author

Antti Karttunen, Nov 06 2017

Keywords

Crossrefs

Programs

  • Mathematica
    Table[DivisorSum[n, 1 &, And[Mod[#, 3] == 2, # != n] &], {n, 105}] (* Michael De Vlieger, Nov 08 2017 *)
  • PARI
    A293896(n) = sumdiv(n,d,(d
    				

Formula

a(n) = A001822(n) - [n == 2 (mod 3)].
G.f.: Sum_{k>=1} x^(6*k-2) / (1 - x^(3*k-1)). - Ilya Gutkovskiy, Apr 14 2021
Sum_{k=1..n} a(k) = n*log(n)/3 + c*n + O(n^(1/3)*log(n)), where c = gamma(2,3) - (2 - gamma)/3 = A256843 - (2 - A001620)/3 = -0.401054... (Smith and Subbarao, 1981). - Amiram Eldar, Nov 25 2023

A218443 a(n) = Sum_{k=0..n} floor(n/(3k+2)).

Original entry on oeis.org

0, 0, 1, 1, 2, 3, 4, 4, 6, 6, 8, 9, 10, 10, 12, 13, 15, 16, 17, 17, 20, 20, 22, 23, 25, 26, 28, 28, 30, 31, 33, 33, 36, 37, 39, 41, 42, 42, 44, 44, 48, 49, 51, 51, 54, 55, 57, 58, 60, 60, 63, 64, 66, 67, 68, 70, 74, 74, 76, 77, 80, 80, 82, 82, 85, 87, 89, 89, 92, 93, 97, 98, 100, 100, 102, 103, 105, 107, 109, 109
Offset: 0

Views

Author

Benoit Cloitre, Oct 28 2012

Keywords

Crossrefs

Partial sums of A001822.

Programs

  • Maple
    N:= 100: # to get a(0)..a(N)
    A001822:= Vector(N+1):
    for m from 2 to N by 3 do
      L:= [seq(i,i=m+1..N+1,m)]:
      A001822[L]:= map(`+`,A001822[L],1)
    od:
    ListTools:-PartialSums(convert(A001822,list)); # Robert Israel, Feb 28 2017
  • Mathematica
    Table[Sum[Floor[n/(3k+2)],{k,0,n}],{n,0,80}] (* Harvey P. Dale, Jun 22 2013 *)
    d[n_] := DivisorSum[n, 1 &, Mod[#, 3] == 2 &]; d[0] = 0; Accumulate@Array[d, 100, 0] (* Amiram Eldar, Nov 25 2023 *)
  • Maxima
    A218443[n]:=sum(floor(n/(3*k+2)),k,0,n)$
    makelist(A218443[n],n,0,80); /* Martin Ettl, Oct 29 2012 */
  • PARI
    a(n)=sum(k=0,n\3,(n\(3*k+2)))
    

Formula

G.f.: Sum_{k>=0} x^(3*k+2)/((1-x^(3*k+2))*(1-x)). - Robert Israel, Feb 28 2017
a(n) = n*log(n)/3 + c*n + O(n^(1/3)*log(n)), where c = gamma(2,3) - (1 - gamma)/3 = A256843 - (1 - A001620)/3 = -0.0677207... (Smith and Subbarao, 1981). - Amiram Eldar, Nov 25 2023
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