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 1-2 of 2 results.

A368551 Decimal expansion of 6*gamma/Pi^2 - 72*zeta'(2)/Pi^4.

Original entry on oeis.org

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

Views

Author

Artur Jasinski, Dec 29 2023

Keywords

Comments

Also the Wolf-Kawalec constant of index 0.
For the Wolf-Kawalec constant of index 1 see A368547.
For the Wolf-Kawalec constant of index 2 see A368568.
Let g(n) be the Wolf-Kawalec constant of index n; then the function
zeta(x)/zeta(2*x) - 6/(Pi^2*(x-1))
has the expansion
Sum_{n>=0} (-1)^n*(g(n)/n!)*(x-1)^n
at x=1.

Examples

			1.0438945157119382974...

Links

Crossrefs

Cf. A000796, A001620, A008683, A073002, A074962, A368547, A368568.

Programs

  • Mathematica
    RealDigits[6 EulerGamma/Pi^2 - 72 Zeta'[2]/Pi^4, 10, 105][[1]]

Formula

Equals (6/Pi^2)*(24*Glaisher - gamma - 2*log(2*Pi)) where Glaisher is A074962.
Equals lim_{x->oo} {(Sum_{n=1..x} abs(mu(n))/n) - 6*log(x)/Pi^2}.

A368568 Decimal expansion of the Wolf-Kawalec constant of index 2.

Original entry on oeis.org

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

Views

Author

Artur Jasinski, Dec 30 2023

Keywords

Comments

For the Wolf-Kawalec constant of index 0 see A368551.
For the Wolf-Kawalec constant of index 1 see A368547.

Examples

			0.3193841204080142492494652...

Links

Crossrefs

Cf. A001620, A073002, A086279, A201994, A368547, A368551.

Programs

  • Mathematica
    RealDigits[Limit[D[D[Zeta[x]/Zeta[2 x] - 6/(Pi^2 (x - 1)), x], x], x -> 1],10,105][[1]]

Formula

Equals 6*(Pi^6*gamma_2 - 3456*(zeta'(2))^3 + 288*Pi^2*zeta'(2)*(gamma*zeta'(2) + 2*zeta''(2)) + 8*Pi^4*(3*gamma_1*zeta(2) - 3*gamma*zeta''(2) - 2*zeta'''(2)))/Pi^8 where gamma_2 is A086279.
Showing 1-2 of 2 results.

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