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.

A196504 Decimal expansion of the least x > 0 satisfying x + tan(x) = 0.

Original entry on oeis.org

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

Views

Author

Clark Kimberling, Oct 03 2011

Keywords

Comments

Let L be the least x > 0 satisfying x + tan(x) = 0.
Then L is also the least x > 0 satisfying x = (sin(x))(sqrt(1+x^2)).
Consequently, for 0 < x < L, for all p > 0, 1/sqrt(1+x^2) - 1/x^p < sin(x) < 1/sqrt(1+x^2) for 0 < x < L.
See A196500-A196503 and A196505 for related constants and inequalities.
The number L also occurs in connection with Du Bois Reymond's constants; see the Finch reference.
For x = L the area of right triangle with vertices (0,0), (x,0) and (x,sin(x)), i.e., the one inscribed into the half-wave curve, is maximal. - Roman Witula, Feb 05 2015

Examples

			L = 2.02875783811043422357697112473471437610838002...
1/L = 0.4929124517549075741877801898222329769156970132...

References

  • Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, page 239.

Links

Crossrefs

Cf. A196505, A196500, A196502.

Programs

  • Mathematica
    Plot[{Sin[x], x/Sqrt[1 + x^2]}, {x, 0, 9}]
t = x /.FindRoot[Sin[x] == x/Sqrt[1 + x^2], {x, .10, 3}, WorkingPrecision -> 100]
RealDigits[t]   (* A196504 *)
1/t
RealDigits[1/t] (* A196505 *)
  • PARI
    solve(x=2,3, sin(x)-x/sqrt(1+x^2)) \\ Charles R Greathouse IV, Feb 11 2025

A196502 Decimal expansion of the least positive x satisfying cos(x)=1/sqrt(1+x^2).

Original entry on oeis.org

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

Views

Author

Clark Kimberling, Oct 03 2011

Keywords

Comments

Let L be the least x>0 satisfying cos(x)=1/sqrt(1+x^2).
Then cos(x) < 1/sqrt(1+x^2) for 0
Consequently,
cos(x) < 1/sqrt(1+x^2) < (1/x)sin(x) for 0
(see A196504). Equivalently,
cos(1/x) < x/sqrt(1+x^2) < x sin(1/x) for x>0.49291...
(see A196505).

Examples

			L=4.9131804394348836888378206685945355668...
1/L=0.203534149076564439697574222397395290289996941...

Links

Crossrefs

Cf. A196500, A196503, A196504.

Programs

  • Mathematica
    Plot[{Cos[x], 1/Sqrt[1 + x^2]}, {x, 0, 8}]
t = x /.FindRoot[1/Sqrt[1 + x^2] == Cos[x], {x, 4, 5}, WorkingPrecision -> 100]
RealDigits[t]   (* A196502 *)
1/t
RealDigits[1/t] (* A196503 *)

A196503 Decimal expansion of greatest x satisfying cos(x)=1/sqrt(1+x^2).

Original entry on oeis.org

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

Views

Author

Clark Kimberling, Oct 03 2011

Keywords

Comments

See A196502 and A196500 for related inequalities.

Examples

			x=0.203534149076564439697574222397395290289996941...

Crossrefs

Cf. A196500, A196502, A196504.

Programs

  • Mathematica
    Plot[{Cos[x], 1/Sqrt[1 + x^2]}, {x, 0, 8}]
t = x /.FindRoot[1/Sqrt[1 + x^2] == Cos[x], {x, 4, 5}, WorkingPrecision -> 100]
RealDigits[t]   (* A196502 *)
1/t
RealDigits[1/t] (* A196503 *)

A196505 Decimal expansion of greatest x>0 satisfying sin(1/x)=1/sqrt(1+x^2).

Original entry on oeis.org

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

Views

Author

Clark Kimberling, Oct 03 2011

Keywords

Comments

Let M be the greatest x>0 satisfying sin(1/x)=1/sqrt(1+x^2). Then sin(1/x) > 1/sqrt(1+x^2) for all x>M=0.4929... See A196500-A196504 for related constants and inequalities.

Examples

			x=0.4929124517549075741877801898222329769156970132...

Crossrefs

Cf. A196500, A196502, A196503.

Programs

  • Mathematica
    Plot[{Sin[x], x/Sqrt[1 + x^2]}, {x, 0, 9}]
Plot[{Sin[1/x], 1/Sqrt[1 + x^2]}, {x, 0.1, 1.0}] (for A196505)
t = x /.FindRoot[Sin[x] == x/Sqrt[1 + x^2], {x, .10, 3}, WorkingPrecision -> 100]
RealDigits[t]   (* A196504 *)
1/t
RealDigits[1/t] (* A196505 *)

A196501 Decimal expansion of least positive x satisfying x+cot(x)=1/x.

Original entry on oeis.org

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

Views

Author

Clark Kimberling, Oct 03 2011

Keywords

Comments

See A196500 for related inequalities.

Examples

			x=2.7437072699922693825611220811203071372042...

Crossrefs

Cf. A196500, A196502, A196504.

Programs

  • Mathematica
    Plot[{Cot[1/x], x, 1/x + Cot[1/x]}, {x, 0.34, 1.0}]
t = x /.FindRoot[1/x + Cot[1/x] == x, {x, .3, .4}, WorkingPrecision -> 100]
RealDigits[t] (* A196500 *)
1/t
RealDigits[%] (* A196501 *)
Showing 1-5 of 5 results.

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