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 1-10 of 10 results.

A196758 Decimal expansion of the number c for which the curve y=1/x is tangent to the curve y=c*sin(x), and 0 < x < 2*Pi.

Original entry on oeis.org

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

Views

Author

Clark Kimberling, Oct 06 2011

Keywords

Examples

			c=0.5495393993551534115219389873253839380900...

Crossrefs

Cf. A196624.

Programs

  • Mathematica
    Plot[{1/x, .55*Sin[x]}, {x, 0, Pi}]
xt = x /. FindRoot[x + Tan[x] == 0, {x, 1.5, 2.5}, WorkingPrecision -> 100]
RealDigits[xt] (* A196504 *)
c = N[1/(xt*Sin[xt]), 100]
RealDigits[c]  (* A196758 *)
slope = -1/xt^2
RealDigits[slope]  (* A196759 *)
  • PARI
    t=solve(x=2,3, sin(x)-x/sqrt(1+x^2)); 1/t/sin(t) \\ Charles R Greathouse IV, Feb 11 2025

A196765 Decimal expansion of the positive number c for which the curve y=c/x is tangent to the curve y=sin(x), and 0 < x < 2*Pi.

Original entry on oeis.org

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

Views

Author

Clark Kimberling, Oct 06 2011

Keywords

Comments

Also, the least local maximum of x*sin(x), which occurs exactly at x = +-A196504, where x = +A196504 is the x-coordinate of this point of tangency of c/x and sin(x) in the first quadrant. There also exists a negative constant d such that d/x and sin(x) are tangent in the fourth quadrant for 0 < x < 2*Pi. - Rick L. Shepherd, Jan 12 2014

Examples

			x=1.8197057411596530462069576803755281456165224784...

Crossrefs

Cf. A196760.

Programs

  • Mathematica
    Plot[{Sin[x], 1/x, 1.82/x}, {x, 0, Pi}]
xt = x /. FindRoot[x + Tan[x] == 0, {x, 1.5, 2.5}, WorkingPrecision -> 100]
RealDigits[xt]   (* A196504 *)
c = N[xt*Sin[xt], 100]
RealDigits[c]    (* A196765 *)
slope = Cos[xt]
RealDigits[slope](* A196766 *)
(* 2nd program for A196765 by Rick L. Shepherd, Jan 12 2014 *)
RealDigits[N[MaxValue[{x*Sin[x], x>1 && x<3}, {x}], 120]]

Extensions

More terms from Rick L. Shepherd, Jan 12 2014

A196500 Decimal expansion of the greatest x satisfying x=1/x+cot(1/x).

Original entry on oeis.org

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

Views

Author

Clark Kimberling, Oct 03 2011

Keywords

Comments

Let B be the greatest x satisfying x=1/x+cot(1/x), so that B=0.364... Then
...
cot(1/x) < x < 1/x+cot(1/x) for all x > B; equivalently,
...
cot(x) < 1/x < x+cot(x) for 0 < x < 1/B = 2.7437....
...
These inequalities and those at A196503 supplement the trigonometric inequalities given in Bullen's dictionary cited below.

Examples

			B=0.364470368591040538004400214637816084912410...
1/B=2.7437072699922693825611220811203071372042...

References

  • P. S. Bullen, A Dictionary of Inequalities, Longman, 1998, pages 250-251.

Links

Crossrefs

Cf. A196501, 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 *)

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 *)

A196759 Decimal expansion of the slope (negative) at the point of tangency of the curves y=1/x and y=c*sin(x), where c is given by A196758.

Original entry on oeis.org

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

Views

Author

Clark Kimberling, Oct 06 2011

Keywords

Examples

			x=-0.242962685095034086912611580795123073012...

Crossrefs

Cf. A196758.

Programs

  • Mathematica
    Plot[{1/x, .55*Sin[x]}, {x, 0, Pi}]
xt = x /. FindRoot[x + Tan[x] == 0, {x, 1.5, 2.5}, WorkingPrecision -> 100]
RealDigits[xt] (* A196504 *)
c = N[1/(xt*Sin[xt]), 100]
RealDigits[c]  (* A196758 *)
slope = -1/xt^2
RealDigits[slope]  (* A196759 *)

A196766 Decimal expansion of the slope (negative) at the point of tangency of the curves y=c/x and y=sin(x), where c is given by A196765.

Original entry on oeis.org

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

Views

Author

Clark Kimberling, Oct 06 2011

Keywords

Examples

			x=-0.44212059295499839133561624405047613618690708...

Crossrefs

Cf. A196765.

Programs

  • Mathematica
    Plot[{Sin[x], 1/x, 1.82/x}, {x, 0, Pi}]
xt = x /. FindRoot[x + Tan[x] == 0, {x, 1.5, 2.5}, WorkingPrecision -> 100]
RealDigits[xt]   (* A196504 *)
c = N[xt*Sin[xt], 100]
RealDigits[c]    (* A196765 *)
slope = Cos[xt]
RealDigits[slope](* A196766 *)

A244257 Decimal expansion of the asymptotic evaluation of the constrained maximum of a certain quadratic form.

Original entry on oeis.org

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

Views

Author

Jean-François Alcover, Jun 24 2014

Keywords

Comments

The quadratic form to maximize is (sum_(k>=1) x(k)/k)^2 + sum_(k>=1) (x(k)/k)^2, subject to the constraint (sum_(k>=1) x(k)^2) <= 1.

Examples

			2.397945586114436337406139378906...

References

  • Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 3.12 Du Bois Reymond's constants, p. 239.

Crossrefs

Cf. A196504.

Programs

  • Mathematica
    xi = x /. FindRoot[x + Tan[x] == 0, {x, 2}, WorkingPrecision -> 100]; RealDigits[(Pi/xi)^2] // First

Formula

(Pi/xi)^2, where xi is the smallest positive solution of the equation x+tan(x)=0.
Showing 1-10 of 10 results.

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