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 1-6 of 6 results.

A103133 Decimal expansion of Dirichlet series L_{-7}(2).

Original entry on oeis.org

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

Views

Author

Eric W. Weisstein, Jan 23 2005

Keywords

Examples

			1.151925470544491047...

Links

Crossrefs

Cf. A326919, A327135.

Programs

  • Mathematica
    (PolyGamma[1, 1/7] + PolyGamma[1, 2/7] - PolyGamma[1, 3/7] + PolyGamma[1, 4/7] - PolyGamma[1, 5/7] - PolyGamma[1, 6/7])/49 // RealDigits[#, 10, 102]& // First

Formula

(Psi(1, 1/7) + Psi(1, 2/7) - Psi(1, 3/7) + Psi(1, 4/7) - Psi(1, 5/7) - Psi(1, 6/7))/49, where Psi(1, x) is the polygamma function of order 1.
Equals Sum_{n>=1} A175629(n)/n^2. - R. J. Mathar, Jan 15 2021
Equals 1/(Product_{p prime == 1, 2 or 4 (mod 7)} (1 - 1/p^2) * Product_{p prime == 3, 5 or 6 (mod 7)} (1 + 1/p^2)). - Amiram Eldar, Dec 17 2023

Extensions

Formula updated by Jean-François Alcover, Apr 01 2015

A329715 Decimal expansion of Sum_{k>=1} Kronecker(8,k)/k^3.

Original entry on oeis.org

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

Views

Author

Jianing Song, Nov 19 2019

Keywords

Comments

Let Chi() be a primitive character modulo d, the so-called Dirichlet L-series L(s,Chi) is the analytic continuation (see the functional equations involving L(s,Chi) in the MathWorld link entitled Dirichlet L-Series) of the sum Sum_{k>=1} Chi(k)/k^s, Re(s)>0 (if d = 1, the sum converges requires Re(s)>1).
If s != 1, we can represent L(s,Chi) in terms of the Hurwitz zeta function by L(s,Chi) = (Sum_{k=1..d} Chi(k)*zeta(s,k/d))/d^s.
L(s,Chi) can also be represented in terms of the polylog function by L(s,Chi) = (Sum_{k=1..d} Chi'(k)*polylog(s,u^k))/(Sum_{k=1..d} Chi'(k)*u^k), where Chi' is the complex conjugate of Chi, u is any primitive d-th root of unity.
If m is a positive integer, we have L(m,Chi) = (Sum_{k=1..d} Chi(k)*polygamma(m-1,k/d))/((-d)^m*(m-1)!).
In this sequence we have Chi = A091337 and s = 3.

Examples

			1 - 1/3^3 - 1/5^3 + 1/7^3 + 1/9^3 - 1/11^3 - 1/13^3 + 1/15^3 + ... = 0.9583804545...

Links

Crossrefs

Cf. A091337.
Decimal expansion of Sum_{k>=1} Kronecker(d,k)/k^3, where d is a fundamental discriminant: A251809 (d=-8), A327135 (d=-7), A153071 (d=-4), A129404 (d=-3), A002117 (d=1), A328723 (d=5), this sequence (d=8), A329716 (d=12).
Decimal expansion of Sum_{k>=1} Kronecker(8,k)/k^s: A196525 (s=1), A328895 (s=2), this sequence (s=3).

Programs

  • Mathematica
    (PolyGamma[2, 1/8] - PolyGamma[2, 3/8] - PolyGamma[2, 5/8] + PolyGamma[2, 7/8])/(-1024) // RealDigits[#, 10, 102] & // First

Formula

Equals (zeta(3,1/8) - zeta(3,3/8) - zeta(3,5/8) + zeta(3,7/8))/512, where zeta(s,a) is the Hurwitz zeta function.
Equals (polylog(3,u) - polylog(3,u^3) - polylog(3,-u) + polylog(3,-u^3))/sqrt(8), where u = sqrt(2)/2 + i*sqrt(2)/2 is an 8th primitive root of unity, i = sqrt(-1).
Equals (polygamma(2,1/8) - polygamma(2,3/8) - polygamma(2,5/8) + polygamma(2,7/8))/(-1024).
Equals 1/(Product_{p prime == 1 or 7 (mod 8)} (1 - 1/p^3) * Product_{p prime == 3 or 5 (mod 8)} (1 + 1/p^3)). - Amiram Eldar, Dec 17 2023

A328723 Decimal expansion of Sum_{k>=1} Kronecker(5,k)/k^3.

Original entry on oeis.org

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

Views

Author

Jianing Song, Nov 19 2019

Keywords

Comments

Let Chi() be a primitive character modulo d, the so-called Dirichlet L-series L(s,Chi) is the analytic continuation (see the functional equations involving L(s,Chi) in the MathWorld link entitled Dirichlet L-Series) of the sum Sum_{k>=1} Chi(k)/k^s, Re(s)>0 (if d = 1, the sum converges requires Re(s)>1).
If s != 1, we can represent L(s,Chi) in terms of the Hurwitz zeta function by L(s,Chi) = (Sum_{k=1..d} Chi(k)*zeta(s,k/d))/d^s.
L(s,Chi) can also be represented in terms of the polylog function by L(s,Chi) = (Sum_{k=1..d} Chi'(k)*polylog(s,u^k))/(Sum_{k=1..d} Chi'(k)*u^k), where Chi' is the complex conjugate of Chi, u is any primitive d-th root of unity.
If m is a positive integer, we have L(m,Chi) = (Sum_{k=1..d} Chi(k)*polygamma(m-1,k/d))/((-d)^m*(m-1)!).
In this sequence we have Chi = A080891 and s = 3.

Examples

			1 - 1/2^3 - 1/3^3 + 1/4^3 + 1/6^3 - 1/7^3 - 1/8^3 + 1/9^3 + ... = 0.8548247666...

Links

Crossrefs

Cf. A080891.
Decimal expansion of Sum_{k>=1} Kronecker(d,k)/k^3, where d is a fundamental discriminant: A251809 (d=-8), A327135 (d=-7), A153071 (d=-4), A129404 (d=-3), A002117 (d=1), this sequence (d=5), A329715 (d=8), A329716 (d=12).
Decimal expansion of Sum_{k>=1} Kronecker(5,k)/k^s: A086466 (s=1), A328717 (s=2), this sequence (s=3).

Programs

  • Mathematica
    (PolyGamma[2, 1/5] - PolyGamma[2, 2/5] - PolyGamma[2, 3/5] + PolyGamma[2, 4/5])/(-250) // RealDigits[#, 10, 102] & // First

Formula

Equals (zeta(3,1/5) - zeta(3,2/5) - zeta(3,3/5) + zeta(3,4/5))/25, where zeta(s,a) is the Hurwitz zeta function.
Equals (polylog(3,u) - polylog(3,u^2) - polylog(3,u^3) + polylog(3,u^4))/sqrt(5), where u = exp(2*Pi*i/5) is a 5th primitive root of unity, i = sqrt(-1).
Equals (polygamma(2,1/5) - polygamma(2,2/5) - polygamma(2,3/5) - polygamma(2,4/5))/(-250).
Equals 1/(Product_{p prime == 1 or 4 (mod 5)} (1 - 1/p^3) * Product_{p prime == 2 or 3 (mod 5)} (1 + 1/p^3)). - Amiram Eldar, Dec 17 2023

A326919 Decimal expansion of Sum_{k>=1} Kronecker(-7,k)/k.

Original entry on oeis.org

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

Views

Author

Jianing Song, Nov 19 2019

Keywords

Comments

Let Chi() be a primitive character modulo d, the so-called Dirichlet L-series L(s,Chi) is the analytic continuation (see the functional equations involving L(s,Chi) in the MathWorld link entitled Dirichlet L-Series) of the sum Sum_{k>=1} Chi(k)/k^s, Re(s)>0 (if d = 1, the sum converges requires Re(s)>1).
If s != 1, we can represent L(s,Chi) in terms of the Hurwitz zeta function by L(s,Chi) = (Sum_{k=1..d} Chi(k)*zeta(s,k/d))/d^s.
L(s,Chi) can also be represented in terms of the polylog function by L(s,Chi) = (Sum_{k=1..d} Chi'(k)*polylog(s,u^k))/(Sum_{k=1..d} Chi'(k)*u^k), where Chi' is the complex conjugate of Chi, u is any primitive d-th root of unity.
If m is a positive integer, we have L(m,Chi) = (Sum_{k=1..d} Chi(k)*polygamma(m-1,k/d))/((-d)^m*(m-1)!).
In this sequence we have Chi = A175629 and s = 1.

Examples

			1 + 1/2 - 1/3 + 1/4 - 1/5 - 1/6 + 1/8 + 1/9 - 1/10 + 1/11 - 1/12 - 1/13 + ... = Pi/sqrt(7) = 1.1874104117...

Links

Crossrefs

Cf. A175629.
Decimal expansion of Sum_{k>=1} Kronecker(d,k)/k, where d is a fundamental discriminant: A093954 (d=-8), this sequence (d=-7), A003881 (d=-4), A073010 (d=-3), A086466 (d=5), A196525 (d=8), A196530 (d=12).
Decimal expansion of Sum_{k>=1} Kronecker(-7,k)/k^s: this sequence (s=1), A103133 (s=2), A327135 (s=3).

Programs

  • Mathematica
    RealDigits[Pi/Sqrt[7], 10, 102] // First
  • PARI
    default(realprecision, 100); Pi/sqrt(7)

Formula

Equals Pi/sqrt(7). This is related to the class number formula: if d<0 is the fundamental discriminant of an imaginary quadratic number field, Chi(k) = Kronecker(d,k), then L(1,Chi) = Sum_{k>=1} Kronecker(d,k)/k = 2*Pi*h(d)/(sqrt(|d|)*w(d)), where h(d) is the class number of K = Q[sqrt(d)], w(d) is the number of elements in K whose norms are 1 (w(d) = 6 if d = -3, 4 if d = -4 and 2 if d < -4). Here d = -7, h(d) = 1, w(d) = 2.
Equals (polylog(1,u) + polylog(1,u^2) - polylog(1,u^3) + polylog(1,u^4) - polylog(1,u^5) - polylog(1,u^6))/sqrt(-7), where u = exp(2*Pi*i/7) is a 7th primitive root of unity, i = sqrt(-1).
Equals (polygamma(0,1/7) + polygamma(0,2/7) - polygamma(0,3/7) + polygamma(0,4/7) - polygamma(0,5/7) - polygamma(0,6/7))/49.
Equals 1/Product_{p prime} (1 - Kronecker(-7,p)/p), where Kronecker(-7,p) = 0 if p = 7, 1 if p == 1, 2 or 4 (mod 7) or -1 if p == 3, 5 or 6 (mod 7). - Amiram Eldar, Dec 17 2023

A329716 Decimal expansion of Sum_{k>=1} Kronecker(12,k)/k^3.

Original entry on oeis.org

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

Views

Author

Jianing Song, Nov 19 2019

Keywords

Comments

Let Chi() be a primitive character modulo d, the so-called Dirichlet L-series L(s,Chi) is the analytic continuation (see the functional equations involving L(s,Chi) in the MathWorld link entitled Dirichlet L-Series) of the sum Sum_{k>=1} Chi(k)/k^s, Re(s)>0 (if d = 1, the sum converges requires Re(s)>1).
If s != 1, we can represent L(s,Chi) in terms of the Hurwitz zeta function by L(s,Chi) = (Sum_{k=1..d} Chi(k)*zeta(s,k/d))/d^s.
L(s,Chi) can also be represented in terms of the polylog function by L(s,Chi) = (Sum_{k=1..d} Chi'(k)*polylog(s,u^k))/(Sum_{k=1..d} Chi'(k)*u^k), where Chi' is the complex conjugate of Chi, u is any primitive d-th root of unity.
If m is a positive integer, we have L(m,Chi) = (Sum_{k=1..d} Chi(k)*polygamma(m-1,k/d))/((-d)^m*(m-1)!).
In this sequence we have Chi = A110161 and s = 3.

Examples

			1 - 1/5^3 - 1/7^3 + 1/11^3 + 1/13^3 - 1/17^3 - 1/19^3 + 1/23^3 + ... = 0.9900400194...

Links

Crossrefs

Cf. A110161.
Decimal expansion of Sum_{k>=1} Kronecker(d,k)/k^3, where d is a fundamental discriminant: A251809 (d=-8), A327135 (d=-7), A153071 (d=-4), A129404 (d=-3), A002117 (d=1), A328723 (d=5), A329715 (d=8), this sequence (d=12).
Decimal expansion of Sum_{k>=1} Kronecker(12,k)/k^s: A196530 (s=1), A258414 (s=2), this sequence (s=3).

Programs

  • Mathematica
    (PolyGamma[2, 1/12] - PolyGamma[2, 5/12] - PolyGamma[2, 7/12] + PolyGamma[2, 11/12])/(-3456) // RealDigits[#, 10, 102] & // First

Formula

Equals (zeta(3,1/12) - zeta(3,5/12) - zeta(3,7/12) + zeta(3,11/12))/1728, where zeta(s,a) is the Hurwitz zeta function.
Equals (polylog(3,u) - polylog(3,u^5) - polylog(3,-u) + polylog(3,-u^5))/sqrt(12), where u = (sqrt(3)+i)/2 is a 12th primitive root of unity, i = sqrt(-1).
Equals (polygamma(2,1/12) - polygamma(2,5/12) - polygamma(2,7/12) + polygamma(2,11/12))/(-3456).
Equals 1/(Product_{p prime == 1 or 11 (mod 12)} (1 - 1/p^3) * Product_{p prime == 5 or 7 (mod 12)} (1 + 1/p^3)). - Amiram Eldar, Dec 17 2023

A105634 Expansion of Sum_{k>0} Kronecker(k,7)*x^k*(1 + x^k)/(1 - x^k)^3.

Original entry on oeis.org

1, 5, 8, 21, 24, 40, 49, 85, 73, 120, 122, 168, 168, 245, 192, 341, 288, 365, 360, 504, 392, 610, 530, 680, 601, 840, 656, 1029, 842, 960, 960, 1365, 976, 1440, 1176, 1533, 1370, 1800, 1344, 2040, 1680, 1960, 1850, 2562, 1752, 2650, 2208, 2728, 2401, 3005
Offset: 1

Views

Author

Michael Somos, Apr 16 2005, Mar 31 2008

Keywords

Examples

			q + 5*q^2 + 8*q^3 + 21*q^4 + 24*q^5 + 40*q^6 + 49*q^7 + 85*q^8 + 73*q^9 + ...

References

  • A. Balog, H. Darmon and K. Ono, Congruence for Fourier coefficients of half-integral weight modular forms and special values of L-functions, pp. 105-128 of Analytic number theory, Vol. 1, Birkhäuser, Boston, 1996, see page 107.
  • Bruce Berndt, Commentary on Ramanujan's Papers, pp. 357-426 of Collected Papers of Srinivasa Ramanujan, Ed. G. H. Hardy et al., AMS Chelsea, 2000. See page 372 (4).

Crossrefs

Cf. A002656, A053724, A138809, A327135.

Programs

  • Mathematica
    f[p_, e_] := If[MemberQ[{1, 2, 4}, Mod[p, 7]], (p^(2*e+2)-1)/(p^2-1), (p^(2*e+2)+(-1)^e)/(p^2+1)]; f[7, e_] := 7^(2*e); a[n_] := Times @@ f @@@ FactorInteger[n]; a[1] = 1; Array[a, 100] (* Amiram Eldar, Sep 04 2023 *)
  • PARI
    {a(n)=local(A,p,e); if(n<2, n==1, A=factor(n); prod(k=1,matsize(A)[1], if(p=A[k,1], e=A[k,2]; if(p==7, p^(2*e), if(kronecker(p,7)==1, (p^(2*e+2)-1)/(p^2-1), (p^(2*e+2)+(-1)^e)/(p^2+1)))))) }
  • PARI
    {a(n)=local(A,B); if(n<1, 0, n--; A=x*O(x^n); polcoeff( if(B=eta(x^7+A), A=eta(x+A); (A*B)^3+8*x*B^7/A), n))}
  • PARI
    {a(n) = if( n<1, 0, sumdiv(n, d, d^2 * kronecker(-7, n / d)))}

Formula

Multiplicative with a(p^e) = p^(2e) if p = 7; (p^(2e+2)-1)/(p^2-1) if p == 1, 2, 4 (mod 7); (p^(2e+2)+(-1)^e)/(p^2+1) if p == 3, 5, 6 (mod 7).
G.f.: Sum_{k>0} Kronecker(k, 7)*x^k*(1+x^k)/(1-x^k)^3.
a(n) = A002656(n) + 8*A053724(n-2).
a(7n) = 49a(n).
G.f. is a period 1 Fourier series which satisfies f(-1 / (7 t)) = 7^(-1/2) (t/i)^3 g(t) where q = exp(2 Pi i t) and g() is g.f. for A138809.
Sum_{k=1..n} a(k) ~ c * n^3 / 3, where c = 32*Pi^3/(343*sqrt(7)) = 1.093343069... (A327135). - Amiram Eldar, Nov 16 2023
Showing 1-6 of 6 results.

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