A257777 -id:A257777 - cp's OEIS frontend

A248618 Decimal expansion of the solution when inverse Gudermannian(x) equals 1.

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

Offset: 0

Robert G. Wilson v, Oct 09 2014

Inverse of A248617.

This is the angle of the unique wedge having its apex at the origin and kissing the exponential curve y=exp(x) on one side, and its inverse logarithmic function y=log(x) on the other side. - Stanislav Sykora, May 31 2015

			0.86576948323965862428960184619184444137967919924876009961184822974244822945841...
The wedge angle in degrees:
49.6049374208547003776513077348112118247866748819092710723979907940346891648208... - _Stanislav Sykora_, May 31 2015

Stanislav Sykora, Table of n, a(n) for n = 0..2000
Wikipedia, Gudermannian function.

Cf. A248617, A257777, A258428.

evalf(arcsin((exp(2)-1)/(exp(2)+1)),100) # Vaclav Kotesovec, Oct 11 2014

RealDigits[ Gudermannian[ 1], 10, 111][[1]]

asin((exp(2)-1)/(exp(2)+1)) \\ Michel Marcus, Oct 11 2014

atan(exp(1))-atan(1/exp(1)) \\ Stanislav Sykora, May 31 2015

2*atan(exp(1))-Pi/2 \\ Charles R Greathouse IV, Jun 02 2015

Equals arcsin((exp(2)-1)/(exp(2)+1)). - Vaclav Kotesovec, Oct 11 2014
Equals atan(e)-atan(1/e) = A257777-A258428. - Stanislav Sykora, May 31 2015
From Amiram Eldar, Apr 07 2022: (Start)
Equals 2*arctan(tanh(1/2)).
Equals Integral_{x=0..1} sech(x) dx. (End)

A257775 Decimal expansion of (e/2)^2.

Original entry on oeis.org

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

Offset: 1

Stanislav Sykora, May 12 2015

The coefficient a of the unique parabola y = a*x^2 which, at some x > 0, kisses the exponential function y = exp(x). The kissing point coordinates are (2,e^2).

			1.847264024732662556807606865143751953295078892637961831021781955630...

Stanislav Sykora, Table of n, a(n) for n = 1..2000

Cf. A001113, A019739, A257776, A257777.

RealDigits[Exp[2]/4, 10, 120][[1]] (* Amiram Eldar, Jun 26 2023 *)

PARI
```
exp(2)/4
```

A257776 Decimal expansion of (e/3)^3.

Original entry on oeis.org

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

Offset: 0

Stanislav Sykora, May 12 2015

The coefficient a of the unique cubic function y=a*x^3 which kisses the exponential function y=exp(x). In general, a function y = c*x^n kisses the exponential at some x > 0 iff the coefficient c equals (e/n)^n. The kissing point is (n, e^n).

			0.743908774932876582997352950169693255443996586613116672014034601...

Stanislav Sykora, Table of n, a(n) for n = 0..2000

Cf. A001113, A019740; A257775 (n=2), A257777 (n=1).

RealDigits[(E/3)^3, 10, 120][[1]] (* Amiram Eldar, May 22 2023 *)

PARI
```
(exp(3)/3)^3
```

A258428 Decimal expansion of arctan(1/e).

Original entry on oeis.org

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

Offset: 0

Stanislav Sykora, May 31 2015

The slope of the unique straight line passing through the origin which kisses the logarithmic function y=log(x), i.e., the angle (in radians) the tangent line subtends with the X axis. The kissing point coordinates are (e,1).

			0.35251342177761899747085992272395350035945275021939640543781203320...
In degrees:
20.1975312895726498111743461325943940876066625590453644638010046029...

Stanislav Sykora, Table of n, a(n) for n = 0..2000

Cf. A001113, A257777, A248618.

RealDigits[ArcTan[Exp[-1]],10,105][[1]] (* Vaclav Kotesovec, Jun 02 2015 *)

PARI
```
atan(1/exp(1))
```

Equals Integral_{x=1..oo} 1/(2*cosh(x)) dx. - Amiram Eldar, Aug 10 2020

A257896 Decimal expansion of (Pi-arctan(e))/(2*Pi).

Original entry on oeis.org

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

Offset: 0

Stanislav Sykora, May 12 2015

Fraction of the horizon obstructed by the exponential curve y=exp(x) when viewed from the origin.

			0.3061042535821462494754842948127622057989073959973482346216694572...

Stanislav Sykora, Table of n, a(n) for n = 0..2000

Cf. A000796, A257777.

RealDigits[(Pi-ArcTan[E])/(2Pi),10,120][[1]] (* Harvey P. Dale, Oct 23 2017 *)

PARI
```
(Pi-atan(exp(1)))/(2*Pi)
```

Equals 1/2 - A257777/(2*Pi).

cp's OEIS Frontend

A248618 Decimal expansion of the solution when inverse Gudermannian(x) equals 1.

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A257775 Decimal expansion of (e/2)^2.

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A257776 Decimal expansion of (e/3)^3.

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A258428 Decimal expansion of arctan(1/e).

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A257896 Decimal expansion of (Pi-arctan(e))/(2*Pi).

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