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 1-5 of 5 results.

A093721 Decimal expansion of Sum_{n>=1} zeta(2n)/(2n)!.

Original entry on oeis.org

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

Views

Author

Eric W. Weisstein, Apr 12 2004

Keywords

Examples

			0.86900199196290899881105480556139568889249484188057785071064577856...

Links

Crossrefs

Cf. A076813, A093720, A269574, A269611.

Programs

  • Maple
    evalf(Sum(cosh(1/n)-1, n=1..infinity), 120); # Vaclav Kotesovec, Mar 04 2016
  • Mathematica
    digits = 105; z[k_] := z[k] = z[k-1] + N[Sum[Zeta[2n]/(2n)!, {n, 2^(k-1) + 1, 2^k}], digits]; z[0] = N[Pi^2/12, digits]; rd[k_] := rd[k] = RealDigits[z[k]][[1]]; rd[0]; rd[k = 1]; While[rd[k] != rd[k-1], k++]; rd[k] (* Jean-François Alcover, Nov 09 2012 *)
  • PARI
    suminf(n=1, zeta(2*n)/(2*n)!) \\ Michel Marcus, Mar 20 2017

Formula

Equals Sum_{k>=1} (cosh(1/k) - 1). - Vaclav Kotesovec, Mar 04 2016

A269720 Decimal expansion of Sum_{n>=1} (Pi/n - sin(Pi/n)).

Original entry on oeis.org

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

Views

Author

Vaclav Kotesovec, Mar 04 2016

Keywords

Examples

			4.096434891501739832220234588626055495928144165119120475644486640639751...

Crossrefs

Cf. A269574, A269611, A093720, A269768.

Programs

  • Maple
    evalf(Sum(Pi/n - sin(Pi/n), n=1..infinity), 120);
  • Mathematica
    RealDigits[NSum[Pi/n - Sin[Pi/n], {n, 1, Infinity}, WorkingPrecision->200, NSumTerms->10000, PrecisionGoal->120, Method->{"NIntegrate", "MaxRecursion"->100}]][[1]]
(* Be aware that N[Sum[Pi/n - Sin[Pi/n], {n, 1, Infinity}], 120] give an incorrect numerical result, only 25 decimal places are correct! *)
  • PARI
    default(realprecision,120); sumpos(n=1, Pi/n - sin(Pi/n))

Formula

Equals Sum_{k>=2} (-1)^k * Pi^(2*k-1) * Zeta(2*k-1) / (2*k-1)!, where Zeta is the Riemann zeta function.

A269768 Decimal expansion of Sum_{n>=2} (-1)^n * zeta(n)/n!.

Original entry on oeis.org

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

Views

Author

Vaclav Kotesovec, Mar 04 2016

Keywords

Examples

			0.659815254349999514863844174352958996077770074088808541384121349320633989...

Links

Crossrefs

Cf. A093720.

Programs

  • Maple
    evalf(Sum(exp(-1/n)-1+1/n, n=1..infinity), 120);
  • Mathematica
    RealDigits[NSum[Exp[-1/n] - 1 + 1/n, {n, 1, Infinity}, WorkingPrecision -> 200, NSumTerms -> 10000, PrecisionGoal -> 120, Method -> {"NIntegrate", "MaxRecursion" -> 100}]][[1]]

Formula

Equals Sum_{k>=1} (exp(-1/k) - 1 + 1/k).
Comment from Velin Yanev, Mar 03 2023 (Start)
Apparently equals 1/2 - Integral_{x=0..oo} (coth(Pi/x)*(sin(x)/x^2 - 1/x) + 1/Pi) dx.
The proposed expression is difficult to evaluate to arbitrary precision.
Maple code: evalf[50](1/2 - Int(coth(Pi/x)*(sin(x)/x^2 - 1/x) + 1/Pi, x = 0 .. infinity));
Mathematica code: 1/2-NIntegrate[Coth[Pi/t] (Sin[t]/t^2-1/t)+1/Pi,{t,0,Infinity},WorkingPrecision->50,MinRecursion->7]
(End)

A375887 Decimal expansion of Product_{n>=2} zeta(n)^n.

Original entry on oeis.org

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

Views

Author

Richard R. Forberg, Sep 01 2024

Keywords

Comments

It is interesting to note that this product is very close in value to 3 * Sum_{n>=2} (zeta(n)^n-1), A375920, where that factor's first 30 digits are: 3.00012312615292744064909403341.

Examples

			9.766825821453289699230682695640792162028987950967280928488833051400227...

Crossrefs

Cf. A375920,(Sum_{n>=2} (zeta(n)^n-1)), A021002 (Product_{n>=2} zeta(n)), A093720 (Sum_{n >= 2} zeta(n)/n!), A013661 (zeta(2)).

Programs

  • Maple
    evalf(Product(Zeta(n)^n, n = 2 .. infinity), 150); # Vaclav Kotesovec, Sep 02 2024
  • Mathematica
    RealDigits[N[Product[Zeta[n]^n, {n, 2, 500}], 150]][[1]]
  • PARI
    prodinf(k = 2, zeta(k)^k) \\ Amiram Eldar, Sep 02 2024

A375920 Decimal expansion of Sum_{n>=2} (zeta(n)^n - 1).

Original entry on oeis.org

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

Views

Author

Richard R. Forberg, Sep 02 2024

Keywords

Comments

It is interesting to note that this sum is very close in value to 1/3 of Product_{n>=2} zeta(n)^n, A375887, where that factor's first 30 digits are: 0.333319653211135001436063576617.

Examples

			3.255474995780369262094368665069015138075282643803397585341859272265720258...

Crossrefs

Cf. A375887 (Product_{n>=2} zeta(n)^n), A021002 (Product_{n>2} zeta(n)), A093720 (Sum_{n>=2} zeta(n)/n!), A013661 (zeta(2)).

Programs

  • Maple
    evalf(Sum(Zeta(n)^n - 1, n = 2 .. infinity), 120); # Vaclav Kotesovec, Sep 02 2024
  • Mathematica
    RealDigits[N[Sum[Zeta[n]^n - 1, {n, 2, 1000}], 150]][[1]]
Showing 1-5 of 5 results.

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