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 1-3 of 3 results.

A173201 Fixed point of the iteration x |--> x - (sin(x) - cos(x)*x - Pi/2)/(sin(x)*x).

Original entry on oeis.org

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

Views

Author

Gerd Lamprecht (gerdlamprecht(AT)googlemail.com), Feb 12 2010

Keywords

Comments

Decimal expansion of psi, the unique solution on (0,Pi) of sin(psi) - psi*cos(psi) = Pi/2, an auxiliary constant used in the Hall-Tenenbaum inequality applied to real multiplicative functions. - Jean-François Alcover, Sep 05 2014

Examples

			A133731 = cos(1.9056957293.../2)*2.

Links

Crossrefs

Cf. A072112, A133731.

Programs

  • Mathematica
    psi = x /. FindRoot[Sin[x] - x*Cos[x] == Pi/2, {x, 2}, WorkingPrecision -> 102]; RealDigits[psi] // First (* Jean-François Alcover, Sep 05 2014 *)
  • PARI
    solve(x=1,2,sin(x)-x*cos(x)-Pi/2) \\ Charles R Greathouse IV, Mar 03 2021

Formula

x := x - (sin(x) - cos(x)*x - Pi/2)/(sin(x)*x).
Equals 2*arccos(A133731/2).

A072113 Continued fraction expansion of Hall and Tenenbaum constant.

Original entry on oeis.org

0, 3, 23, 1, 1, 16, 1, 2, 1, 8, 1, 274, 3, 1, 5, 1, 2, 1, 16, 1, 3, 3, 2, 1, 4, 1, 1, 2, 1, 2, 1, 2, 1, 1, 2, 16, 3, 3, 2, 1, 1, 1, 2, 69, 121, 1, 5, 1, 2, 1, 2, 1, 1, 1, 2, 1, 12, 4, 1, 1, 1, 1, 2, 1, 2, 3, 3, 1, 3, 2, 4, 1, 7, 1, 16, 2, 4, 1, 2, 7, 2, 3, 1, 3, 2, 1, 1, 1, 1, 2, 1, 1, 3, 1, 1, 3, 2, 1
Offset: 0

Views

Author

Benoit Cloitre, Jun 19 2002

Keywords

Comments

For any multiplicative function g with values -1<= g(k) <= 1, for any real x >=2, Sum( i<= x, g(i) ) << x * exp{ -K * Sum( p<=x, (1-g(p))/p ) } and K is the optimal constant satisfying this inequality ( Hall and Tenenbaum, 1991).

References

  • G. Tenenbaum, Introduction à la théorie analytique et probabiliste des nombres, p. 348, Publications de l'Institut Cartan, 1990.

Crossrefs

Cf. A072112 (decimal expansion).

Programs

  • PARI
    \p200;
contfrac(cos(solve(X=0,2*Pi,sin(X)+(Pi-X)*cos(X)-Pi/2)))

Formula

K = cos(S) = 0.3287... where S it the root 0< S < 2Pi of sin(S)+(Pi-S)*cos(S) = Pi/2.

Extensions

Offset changed by Andrew Howroyd, Jul 06 2024

A173571 Decimal expansion of constant related to Goat Problem, Donkey Problem, Tenenbaum and A173201.

Original entry on oeis.org

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

Views

Author

Gerd Lamprecht (gerdlamprecht(AT)googlemail.com), Feb 22 2010

Keywords

Examples

			0.9528478646... = A075838 =(0.9444433782055...+PI/2-asin(0.9444433782055...))/(A133731^2);

Links

Formula

x = sqrt(1-A072112^2) = sqrt(1-(1-A133731^2/2)^2); A075838 =(x+PI/2-asin(x))/(A133731^2); with A072112=1-A133731^2/2; A133731=cos(A173201/2)*2;
Showing 1-3 of 3 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.