A339606 -id:A339606 - 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.

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

A364449 Lexicographically earliest sequence where n is banned for n^3 terms after each appearance.

Original entry on oeis.org

1, 2, 1, 3, 1, 4, 1, 5, 1, 6, 1, 2, 1, 7, 1, 8, 1, 9, 1, 10, 1, 2, 1, 11, 1, 12, 1, 13, 1, 14, 1, 2, 1, 3, 1, 15, 1, 16, 1, 17, 1, 2, 1, 18, 1, 19, 1, 20, 1, 21, 1, 2, 1, 22, 1, 23, 1, 24, 1, 25, 1, 2, 1, 3, 1, 26, 1, 27, 1, 28, 1, 2, 1, 4, 1, 29, 1, 30, 1, 31, 1, 2, 1, 32, 1, 33, 1, 34, 1, 35

Offset: 1

Rok Cestnik, Jul 25 2023

nonn

Sequence is unbounded. The fastest branch grows asymptotically linearly: limsup a(n)/n > 1-Sum_{n>0} 1/(n^3+1) = 1-A339606 = 0.313496...

If banning for n terms (A364447), or n^2 terms (A364448), the sequence is eventually periodic.

			a(n)   ban 1  2  3  4  5  6  7  ...
 1         |  |  |  |  |  |  |
 2         x  |  |  |  |  |  |
 1         |  x  |  |  |  |  |
 3         x  x  |  |  |  |  |
 1         |  x  x  |  |  |  |
 4         x  x  x  |  |  |  |
 1         |  x  x  x  |  |  |
 5         x  x  x  x  |  |  |
 1         |  x  x  x  x  |  |
 6         x  x  x  x  x  |  |
 1         |  |  x  x  x  x  |
 2         x  |  x  x  x  x  |
 1         |  x  x  x  x  x  |
 7         x  x  x  x  x  x  |
 1         |  x  x  x  x  x  x
 .
 .
 .

Cf. A364447, A364448, A339606.

a = []
ban = [0 for n in range(500)]
for i in range(1000):
    can = ban.index(0,1)
    ban = [max(b-1,0) for b in ban]
    a.append(can)
    ban[can] = can**3

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.

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

A359944 Number of divisors d of n such that d-1 is a cube.

Original entry on oeis.org

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

Offset: 1

Seiichi Manyama, Jan 19 2023

nonn

The Cartesian equation for the Folium of Descartes is given as x^3 + y^3 = 3*k*x*y. If we set 3*k = n, then a(n)-1 is the number of integer solutions such that x,y > 0 and y >= x. Let d = m^3+1 be a divisor of n, then x = 3*k*m/(m^3+1); y = 3*k*m^2/(m^3+1) is a solution. - Thomas Scheuerle, Aug 07 2024

Cf. A001093, A061704, A339606.

a[n_] := DivisorSum[n, 1 &, IntegerQ[Surd[#-1, 3]] &]; Array[a, 100] (* Amiram Eldar, Aug 09 2023 *)

PARI
```
a(n) = sumdiv(n, d, ispower(d-1, 3));
```

G.f.: Sum_{k>=0} x^(k^3+1)/(1 - x^(k^3+1)).

Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Sum_{k>=0} 1/(k^3+1) = 1 + A339606 = 1.686503... . - Amiram Eldar, Jan 01 2024

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

A371945 Decimal expansion of Sum_{k>=0} (-1)^k / (k^3 + 1).

Original entry on oeis.org

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

Offset: 0

Clark Kimberling, Apr 24 2024

			0.58569886194979188645332704695918615397536302...

Cf. A339606.

s = Chop[N[Sum[(-1)^k/(k^3 + 1), {k, 0, Infinity}], 120]]
First[RealDigits[s]]

cp's OEIS Frontend

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

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

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A364449 Lexicographically earliest sequence where n is banned for n^3 terms after each appearance.

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

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

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A359944 Number of divisors d of n such that d-1 is a cube.

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

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

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