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

A339135 Decimal expansion of J = 2*log(2)/3 - Re(Psi(1/2 + i*sqrt(3)/2)), where Psi is the digamma function and i=sqrt(-1).

Original entry on oeis.org

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

Views

Author

Artur Jasinski, Nov 25 2020

Keywords

Comments

Generally in the literature there is no explicit formula for the real part of the function Psi(x + i*y) when y != 0.
Up to now there is no explicit formula expressing the constant J in terms of other mathematical constants.

Examples

			J = 0.677024679102993347...

Crossrefs

Cf. A097663, A097663-A097676, A268046, A268086, A257958, A257959, A338815, A339237.

Programs

  • Maple
    evalf(1 + 2*log(2)/3 - Psi(0, 5/2 - sqrt(3)*I/2)/2 - Psi(0, 5/2 + sqrt(3)*I/2)/2, 100); # Vaclav Kotesovec, Nov 26 2020
  • Mathematica
    RealDigits[N[Re[2 Log[2]/3 - PolyGamma[0, 1/2 + I Sqrt[3]/2]], 110]][[1]]
Chop[N[1 + 2*Log[2]/3 - PolyGamma[0, 5/2 - I*Sqrt[3]/2]/2 - PolyGamma[0, 5/2 + I*Sqrt[3]/2]/2, 120]] (* Vaclav Kotesovec, Nov 26 2020 *)
  • PARI
    2*log(2)/3 - real(psi(1/2 + I*sqrt(3)/2)) \\ Michel Marcus, Nov 25 2020

Formula

J = -log(2)/3 - (1/2)*Pi/cosh(Pi*sqrt(3)/2) - Re(Psi(1/4 + i*sqrt(3)/4)).
J = -log(2)/3 + (1/2)*Pi/cosh(Pi*sqrt(3)/2) - Re(Psi(3/4 + i*sqrt(3)/4)).
J = 3 + gamma + (2/3)*log(2) - (1/2)* sqrt(3)*Pi*tanh(Pi*sqrt(3)/2) - 3*(Sum_{n>=1} zeta(3*n)-1), where zeta is Riemann zeta function and gamma is Euler gamma constant see A001620.
J = -(1/2) + gamma + (2/3)*log(2) + (3/2)*(Sum_{n>=1} zeta(3*n+1)-1).
J = -1 + gamma + (2/3)*log(2) + (1/2)*sqrt(3)*Pi*tanh(Pi*sqrt(3)/2) - 3*(Sum_{n>=0} zeta(3*n+2)-1).
J = -(3/8) + gamma + (2/3)*log(2) + (3/2)*(Sum_{n>=1} zeta(6*n+1)-1).
J = 1/4 + gamma + (2/3)*log(2) - 3*(Sum_{n>=0} zeta(6*n+3)-1).
J = -(1/4) + gamma + (2/3)*log(2) - 3 (Sum_{n>=0} zeta(6*n+5)-1).
J = (11/12 - (1/4)*i*sqrt(3))*Psi(1/2 + i*sqrt(3)/2) + (-(5/4) + (1/4)*i*sqrt(3))*Psi(-(1/2) - i*sqrt(3)/2) + (-(17/24) + (1/8)*i*sqrt(3))* Psi(1/4 + i*sqrt(3)/4) + (3/8 - (1/8)*i*sqrt(3))*Psi(-(1/4) - i*sqrt(3)/4) + (-(17/24) + (1/8)*i*sqrt(3))*Psi(3/4 + i*sqrt(3)/4) + (3/8 - (1/8)*i*sqrt(3))*Psi(-(3/4) - i*sqrt(3)/4).
J = 2*log(2)/3 - Integral_{t=0..infinity} cosh(t)/t - sinh(t)/t - (cos(sqrt(3)*t)*cosh(t/2))/(1 - cosh(t) + sinh(t)) + (cos(sqrt(3)*t)*sinh(t/2))/(1 - cosh(t) + sinh(t)).
J = gamma + (1/6)*Sum_{t>=1} (6*t^3-4*t^2-4*t-1)/(t*(t+1)*(2t+1)*(t^2+t+1)).
Equals 1 + 2*log(2)/3 - Psi(0, 5/2 - i*sqrt(3)/2)/2 - Psi(0, 5/2 + i*sqrt(3)/2)/2. - Vaclav Kotesovec, Nov 26 2020

A339604 Decimal expansion of Sum_{k>=1} (zeta(3*k)-1).

Original entry on oeis.org

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

Views

Author

Artur Jasinski, Dec 09 2020

Keywords

Comments

For additional comments and generalization see attached text file.

Examples

			0.221689395109267...

Links

Crossrefs

Cf. A024006, A256919, A338815, A339135, A339529, A339530, A339605, A339606.

Programs

  • Mathematica
    RealDigits[Chop[N[Sum[Zeta[3 n] - 1, {n, 1, Infinity}], 105]]][[1]]
  • PARI
    suminf(k=1, zeta(3*k)-1) \\ Michel Marcus, Dec 09 2020

Formula

Equals Sum_{k>=2} 1/(k^3-1).
Equals 1 + gamma/3 + (1/3)*Re(Psi(1/2 + i*sqrt(3)/2)) - sqrt(3)*Pi*tanh(sqrt(3)*Pi/2)/6, where Psi is the digamma function, gamma is the Euler-Mascheroni constant (see A001620), and i=sqrt(-1).
Equals 1 + gamma/3 - (1/3)*A339135 + 2*log(2)/9 - sqrt(3)*Pi*tanh(sqrt(3)*Pi/2)/6.
Equals 7/6 - Pi*tanh(Pi*sqrt(3)/2)/(2*sqrt(3)) - A339605/2.
Equals 4/3 - Pi*tanh(Pi*sqrt(3)/2)/sqrt(3) + A339606.
Equals 1 - A339605 - A339606.

A339606 Decimal expansion of Sum_{k>=0} (zeta(3*k+2)-1).

Original entry on oeis.org

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

Views

Author

Artur Jasinski, Dec 09 2020

Keywords

Examples

			0.6865033423386238859646...

Crossrefs

Cf. A024006, A256919, A338815, A339135, A339529, A339530, A339604, A339605.

Programs

  • Maple
    evalf(Re(sum(1/(k^3+1), k=1..infinity)), 120);  # Alois P. Heinz, Dec 12 2020
  • Mathematica
    RealDigits[Chop[N[Sum[Zeta[3 n + 2] - 1, {n, 0, Infinity}], 105]]][[1]]
  • PARI
    suminf(k=0, zeta(3*k+2)-1) \\ Michel Marcus, Dec 09 2020

Formula

Equals Sum_{k>=1} 1/(k^3 + 1).
Equals -1/3 + gamma/3 + (1/3)*Re(Psi(1/2 + i*sqrt(3)/2)) + sqrt(3)*Pi*tanh(sqrt(3)*Pi/2)/6, where Psi is digamma function, gamma is Euler-Mascheroni constant (see A001620), and i=sqrt(-1).
Equals -1/3 + gamma/3 - (1/3)*A339135 + 2*log(2)/9 + sqrt(3)*Pi*tanh(sqrt(3)*Pi/2)/6.
Equals 1 - A339605 - A339604.
Equals 1/2 + Sum_{k>=1} (-1)^(k+1) * (zeta(3*k)-1). - Amiram Eldar, Jan 07 2024

A339605 Decimal expansion of Sum_{k>=1} (zeta(3*k+1) - 1).

Original entry on oeis.org

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

Views

Author

Artur Jasinski, Dec 09 2020

Keywords

Examples

			0.09180726255210907564327636663...

Crossrefs

Cf. A256919, A338815, A339135, A339529, A339530, A339604, A339606.

Programs

  • Mathematica
    Join[{0}, RealDigits[Chop[N[Sum[Zeta[3 n + 1] - 1, {n, 1, Infinity}], 105]]][[1]]]
  • PARI
    suminf(k=1, zeta(3*k+1)-1) \\ Michel Marcus, Dec 09 2020

Formula

Equals Sum_{k>=2} (k^5 - 3*k^4 + k^3 - k^2 + k - 1)/(k*(k^6 - 1)).
Equals 1/3 - 2*gamma/3 - (2/3)*Re(Psi(1/2 + i*sqrt(3)/2)), where Psi is the digamma function, gamma is the Euler-Mascheroni constant (see A001620), and i=sqrt(-1).
Equals 1/3 - 2*gamma/3 + 2*A339135/3 - 4*log(2)/9.
Equals 1 - A339604 - A339606.
Equals Sum_{k>=2} 1/(k*(k^3 - 1)). - Vaclav Kotesovec, Dec 24 2020

A338858 Decimal expansion of Sum_{k>=0} (zeta(4*k+3)-1).

Original entry on oeis.org

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

Views

Author

Artur Jasinski, Dec 24 2020

Keywords

Comments

For additional comments and generalization see A339604.

Examples

			0.2109329927620049189391952864...

Crossrefs

Cf. A256919 (4*k), A339097 (4*k+1), A339083 (4k+2).
Cf. A024006, A338815, A339135, A339529, A339530, A339604, A339605, A339606.

Programs

  • Mathematica
    RealDigits[N[Re[Sum[Zeta[4 n + 3] - 1, {n, 0, Infinity}]], 105]][[1]]
  • PARI
    suminf(k=0, zeta(4*k+3)-1) \\ Michel Marcus, Dec 24 2020

Formula

Equals Sum_{k>=2} k/(k^4-1).
Equals -1/8 + gamma/2 + Re(Psi(i))/2, where Psi is the digamma function, gamma is the Euler-Mascheroni constant (see A001620), and i=sqrt(-1).
Equals -1/8 + Re(H(I))/2, where H is the harmonic number function.

A339083 Decimal expansion of Sum_{k>=0} (zeta(4*k+2)-1).

Original entry on oeis.org

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

Views

Author

Artur Jasinski, Dec 24 2020

Keywords

Comments

Sum_{k>=1} zeta(4*k)-1 see A256919.
Sum_{k>=1} zeta(4*k+1)-1 see A339097.
Sum_{k>=0} zeta(4*k+3)-1 see A338858.
For additional comments and generalization see A339604.

Examples

			0.663337023734290587067025397375...

Crossrefs

Cf. A024006, A256919, A338815, A338858, A339097, A339135, A339529, A339530, A339604, A339605, A339605.

Programs

  • Mathematica
    RealDigits[N[Sum[Zeta[4 n + 2] - 1, {n, 0, Infinity}], 105]][[1]]
  • PARI
    suminf(k=0, zeta(4*k+2)-1) \\ Michel Marcus, Dec 24 2020

Formula

Equals Sum_{k>=2} k^2/(k^4-1).
Equals -1/8 + Pi*coth(Pi)/4 = -1/8 + A338815 = 3/4 - A256919.

Extensions

a(104) corrected and more terms from Georg Fischer, Jun 06 2024

A339097 Decimal expansion of Sum_{k>=1} zeta(4*k+1)-1.

Original entry on oeis.org

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

Views

Author

Artur Jasinski, Dec 24 2020

Keywords

Examples

			0.0390670072379950810608...

Crossrefs

Cf. A256919 (4*k), A339083 (4*k+2), A338858 (4k+3).
Cf. also A338815, A339135, A339529, A339530, A339604, A339605, A339606.

Programs

  • Mathematica
    Join[{0},RealDigits[N[Re[Sum[Zeta[4 n + 1] - 1, {n, 1, Infinity}]], 105]][[1]]]
  • PARI
    suminf(k=1, zeta(4*k+1)-1) \\ Michel Marcus, Dec 24 2020

Formula

Equals Sum_{k>=2} (k^3 -3*k^2 + k - 2)/(k^5 - k).
Equals 3/8 - gamma/2 - Re(Psi(i))/2, where Psi is the digamma function, gamma is the Euler-Mascheroni constant (see A001620), and i=sqrt(-1).
Equals 3/8 - Re(H(I))/2, where H is the harmonic number function.
Equals 1/4 - A338858.
Equals Sum_{k>=2} 1/(k*(k^4 - 1)). - Vaclav Kotesovec, Dec 24 2020

A338851 Constant T such that Sum_{n>=1} zeta(T*n)=1.

Original entry on oeis.org

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

Views

Author

Artur Jasinski, Nov 12 2020

Keywords

Examples

			1.82024771787000913066...

Crossrefs

Cf. A021002, A338815.

Programs

  • Mathematica
    (*convergence test*)
Do[Print[FindRoot[Sum[Zeta[c n] - 1, {n, 1, m}] == 1, {c, 2},
   WorkingPrecision -> 200]], {m, 50, 450, 50}] (* Krzysztof Maslanka *)

A339801 Decimal expansion of the real part of harmonic number H(1/2 + i*sqrt(3)/2), where i=sqrt(-1).

Original entry on oeis.org

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

Views

Author

Artur Jasinski, Dec 17 2020

Keywords

Comments

For imaginary part see A339802.
For real b, Im(Psi(1/2 + b*i)) = Pi*tanh(Pi*b)/2, but no such closed formula is known for the real part (see Wikipedia link). - Vaclav Kotesovec, Dec 19 2020

Examples

			0.862289106171836386535085450...

Links

Crossrefs

Cf. A256919, A338815, A339135, A339529, A339530, A339604, A339605, A339606, A339802.

Programs

  • Mathematica
    RealDigits[N[Re[HarmonicNumber[1/2 + I Sqrt[3]/2]], 105]][[1]]

Formula

Equals 1/2 + gamma + Re(Psi(1/2 + i*sqrt(3)/2)), where gamma is the Euler-Mascheroni constant (see A001620) and Psi is the digamma function.
Equals -1/2 + 3*A339604 + 3*A339606.
Equals Re((1 + i*sqrt(3))*Sum_{k>=0} 1/((1 + k)*(3 + i*sqrt(3) + 2*k))).

A339802 Decimal expansion of the imaginary part of harmonic number H(1/2 + i*sqrt(3)/2) where i=sqrt(-1).

Original entry on oeis.org

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

Views

Author

Artur Jasinski, Dec 17 2020

Keywords

Comments

For real part of H(1/2 + i*sqrt(3)/2) see A339801.

Examples

			0.691215820928755403365848...

Crossrefs

Cf. A256919, A338815, A339135, A339529, A339530, A339605, A339605, A339606, A339801.

Programs

  • Mathematica
    RealDigits[N[Im[HarmonicNumber[1/2 + I Sqrt[3]/2]], 105]][[1]]

Formula

Equals (Pi/2)*tanh(Pi*sqrt(3)/2) - sqrt(3)/2.
Equals Im(Psi(3/2 + i*sqrt(3)/2)).
Equals -sqrt(3)/2 + Im(Psi(1/2 + i*sqrt(3)/2)).
Equals Im((1 + i*sqrt(3))*Sum_{k>=0} 1/((1 + k)*(3 + i*sqrt(3) + 2*k))).
Showing 1-10 of 11 results. Next

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