A339135 -id:A339135 - cp's OEIS frontend

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

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

Artur Jasinski, Dec 09 2020

For additional comments and generalization see attached text file.

			0.221689395109267...

Artur Jasinski, Sums of zeta(p*n+q)-1
T. J. Stieltjes, Table des valeurs des sommes Sk, Acta Math., Volume 10 (1887), 299-302.

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

RealDigits[Chop[N[Sum[Zeta[3 n] - 1, {n, 1, Infinity}], 105]]][[1]]

suminf(k=1, zeta(3*k)-1) \\ Michel Marcus, Dec 09 2020

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

Artur Jasinski, Dec 09 2020

			0.6865033423386238859646...

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

evalf(Re(sum(1/(k^3+1), k=1..infinity)), 120);  # Alois P. Heinz, Dec 12 2020

RealDigits[Chop[N[Sum[Zeta[3 n + 2] - 1, {n, 0, Infinity}], 105]]][[1]]

suminf(k=0, zeta(3*k+2)-1) \\ Michel Marcus, Dec 09 2020

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

Artur Jasinski, Dec 09 2020

			0.09180726255210907564327636663...

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

Join[{0}, RealDigits[Chop[N[Sum[Zeta[3 n + 1] - 1, {n, 1, Infinity}], 105]]][[1]]]

suminf(k=1, zeta(3*k+1)-1) \\ Michel Marcus, Dec 09 2020

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

Artur Jasinski, Dec 24 2020

For additional comments and generalization see A339604.

			0.2109329927620049189391952864...

Cf. A256919 (4*k), A339097 (4*k+1), A339083 (4k+2).

Cf. A024006, A338815, A339135, A339529, A339530, A339604, A339605, A339606.

RealDigits[N[Re[Sum[Zeta[4 n + 3] - 1, {n, 0, Infinity}]], 105]][[1]]

suminf(k=0, zeta(4*k+3)-1) \\ Michel Marcus, Dec 24 2020

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

Artur Jasinski, Dec 24 2020

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.

			0.663337023734290587067025397375...

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

RealDigits[N[Sum[Zeta[4 n + 2] - 1, {n, 0, Infinity}], 105]][[1]]

suminf(k=0, zeta(4*k+2)-1) \\ Michel Marcus, Dec 24 2020

Equals Sum_{k>=2} k^2/(k^4-1).

Equals -1/8 + Pi*coth(Pi)/4 = -1/8 + A338815 = 3/4 - A256919.

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

Artur Jasinski, Dec 24 2020

			0.0390670072379950810608...

Cf. A256919 (4*k), A339083 (4*k+2), A338858 (4k+3).

Cf. also A338815, A339135, A339529, A339530, A339604, A339605, A339606.

Join[{0},RealDigits[N[Re[Sum[Zeta[4 n + 1] - 1, {n, 1, Infinity}]], 105]][[1]]]

suminf(k=1, zeta(4*k+1)-1) \\ Michel Marcus, Dec 24 2020

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

A339237 Decimal expansion of K = Sum_{m>=0} 1/(1 + 2m + 4m^2).

Original entry on oeis.org

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

Offset: 1

Artur Jasinski, Nov 28 2020

This constant K and the constant J = A339135 allow the expression of the real and imaginary parts of:
Psi(1/4 + i*sqrt(3)/4) = - J - log(2)/3 - (1/2)*Pi/cosh(Pi*sqrt(3)/2) + i*sqrt(3)*K;
Psi(-1/4 + i*sqrt(3)/4) = 1 - J - log(2)/3 + (1/2)*Pi/cosh(Pi*sqrt(3)/2) + i*(sqrt(3) - sqrt(3)*K + Pi*tanh(Pi*sqrt(3)/2));
Psi(3/4 + i*sqrt(3)/4)= - J - i*sqrt(3)*k - log(2)/3 + (1/2)*Pi/cosh(Pi*sqrt(3)/4) + i*Pi*tanh(Pi*sqrt(3)/2).
Psi(-3/4 + i*sqrt(3)/4) = 1 - J - log(2)/3 - (1/2)*Pi/cosh(Pi*sqrt(3)/2) + i*(sqrt(3)/3 + sqrt(3)*K).
where Psi is the digamma function and i=sqrt(-1).

			1.27972874228189683364727570150763067226260...

Cf. A054569 (terms), A339135.

K:= Re(sum(1/(1+2*n+4*n^2), n=0..infinity)):
evalf(K, 120);  # Alois P. Heinz, Dec 06 2020

RealDigits[N[Re[Sum[1/(1 + 2*n + 4*n^2), {n, 0, Infinity}]], 110]][[1]]

sumpos(n=0, 1/(1+2*n+4*n^2)) \\ Michel Marcus, Nov 28 2020

Equals -i/(2*sqrt(3)) * (Psi(1/4 + i*sqrt(3)/4) - Psi(1/4 - i*sqrt(3)/4)).

Equals Pi*sqrt(3)*tanh(Pi*sqrt(3)/2)/3 - Sum_{m>=0} 1/(3 + 6*m + 4*m^2).

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

Artur Jasinski, Dec 17 2020

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

			0.862289106171836386535085450...

Wikipedia, Digamma function

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

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

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

Artur Jasinski, Dec 17 2020

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

			0.691215820928755403365848...

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

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

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))).

A351432 Decimal expansion of sqrt(3)/2 + Pitanh(Pisqrt(3)/2)/2.

Original entry on oeis.org

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

Offset: 1

Artur Jasinski, Feb 11 2022

Imaginary part of psi(-1/2 + i*sqrt(3)/2) where psi is the digamma function.

			2.423266628497632696893294495...

Cf. A024006, A100554, A113319, A256919, A259171, A338815, A339135, A339529, A339530, A339604, A339605, A339606, A347292.

RealDigits[N[Sqrt[3]/2 + 1/2 Pi Tanh[Sqrt[3] Pi/2], 105]][[1]]

imag(psi(-1/2+I*sqrt(3)/2)) \\ Michel Marcus, Feb 11 2022

Equals sqrt(3)*(1 - gamma/3 - Re(psi(-1/2 + i*sqrt(3)/2))/3 + A339606).

Last two digits corrected by Georg Fischer, May 15 2024

cp's OEIS Frontend

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

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

Views

Author

Keywords

Examples

Crossrefs

Programs

Maple

Mathematica

PARI

Formula

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

Views

Author

Keywords

Examples

Crossrefs

Programs

Mathematica

PARI

Formula

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

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

PARI

Formula

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

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

PARI

Formula

Extensions

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

Views

Author

Keywords

Examples

Crossrefs

Programs

Mathematica

PARI

Formula

A339237 Decimal expansion of K = Sum_{m>=0} 1/(1 + 2*m + 4*m^2).

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Maple

Mathematica

PARI

Formula

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

A339237 Decimal expansion of K = Sum_{m>=0} 1/(1 + 2m + 4m^2).

A351432 Decimal expansion of sqrt(3)/2 + Pitanh(Pisqrt(3)/2)/2.