Michal Paulovic - cp's OEIS frontend

A370393 Decimal expansion of the area of a unit heptadecagon (17-gon).

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

Offset: 2

Michal Paulovic, Feb 17 2024

This constant multiplied by the square of the side length of a regular heptadecagon equals the area of that heptadecagon.
17^2 divided by this constant equals 68 * tan(Pi/17) = 12.71140300... which is the perimeter and the area of an equable heptadecagon with its side length 4 * tan(Pi/17) = 0.74772958... .
An equable rectangle with its perimeter and area = 17 has side lengths:
  a = s^2/8 = (17 - sqrt(17)) / 4 = (17 - A010473) / 4 = 3.21922359...
  b = 136/s^2 = (17 + sqrt(17)) / 4 = (17 + A010473) / 4 = 5.28077640...
  where s is the parameter from the formula mentioned below.

			22.7354918984165514...

Eric Weisstein's World of Mathematics, Heptadecagon.
Wikipedia, Heptadecagon.

Cf. A007450, A010473, A019684 (Pi/17), A210644 (cos(2*Pi/17)), A210649, A228787, A241243, A329592, A343061.

Maple
```
evalf(17 / (4 * tan(Pi/17)), 100);
```

RealDigits[17 / (4 * Tan[Pi/17]), 10, 100][[1]]

PARI
```
17 / (4 * tan(Pi/17))
```

Equals 17 / (4 * tan(Pi/17)) = 17 / (4 * A343061).
Equals 1 / (4 * A007450 * A343061).
Equals 17 * cos(Pi/17) / (4 * sin(Pi/17)).
Equals 17 * A210649 / (4 * A241243).
Equals 17 * A210649 / (2 * A228787).
Equals 17 * cot(Pi/17) / 4.
Equals 17 * sqrt(4 / (s^2 - 2 * s - 4 * sqrt(17 + 3 * sqrt(17) - s - sqrt(17) * 16/s)) - 1/16) where s = sqrt(34 - 2 * sqrt(17)) = 4 * A329592.
The minimal polynomial is 4294967296*x^16 - 3103113871360*x^14 + 510054948143104*x^12 - 28954726431195136*x^10 + 653743432704327680*x^8 - 6011468019822067712*x^6 + 20881180982314634240*x^4 - 21552361799603318912*x^2 + 2862423051509815793.

A364966 Decimal expansion of the solution to exp(-x^2) = x.

Original entry on oeis.org

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

Offset: 0

Michal Paulovic, Aug 14 2023

Fixed point of Gaussian function.

			0.6529186404192047...

Eric Weisstein's World of Mathematics, Gaussian Function.
Eric Weisstein's World of Mathematics, Lambert W-Function.
Wikipedia, Gaussian function.
Wikipedia, Lambert W function.
Index entries for transcendental numbers

Cf. A196515, A202498, A299624.

Digits:=105: evalf(sqrt(LambertW(2)/2));

RealDigits[Sqrt[ProductLog[2]/2], 10, 105][[1]]

default(realprecision, 105); sqrt(lambertw(2)/2)

Equals sqrt(LambertW(2)/2).
Equals sqrt(A196515/2).
Equals sqrt(A202498).
Equals sqrt(A299624)/2.

A364931 Decimal expansion of the solution to 1 / (1 + e^(-x)) = x.

Original entry on oeis.org

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

Offset: 0

Michal Paulovic, Aug 13 2023

Fixed point of sigmoid (standard logistic function) as well as its inverse (logit).

			0.6590460684074066...

Eric Weisstein's World of Mathematics, Sigmoid Function.
Wikipedia, Logistic Function.

Digits:=100: fsolve(1/(1+exp(-x))-x, x);

RealDigits[FindRoot[1/(1+Exp[-x])-x, {x, 0.6}, WorkingPrecision -> 100][[1, 2]], 10, 100][[1]]

default(realprecision, 100); solve(x=0.4,0.7,1/(1+exp(-x))-x)

A364521 Decimal expansion of the solution to Ei(x) = x.

Original entry on oeis.org

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

Offset: 0

Michal Paulovic, Aug 15 2023

Fixed point of exponential integral.

			0.5276123472017420...

Eric Weisstein's World of Mathematics, Exponential Integral.
Wikipedia, Exponential integral.

Cf. A091723, A091725, A099285.

Maple
```
Digits:=101: fsolve(Ei(1,x)-x, x);
```

RealDigits[FindRoot[ExpIntegralE[1, x] - x, {x, 0.5}, WorkingPrecision -> 101][[1, 2]], 10, 101][[1]]

default(realprecision, 101); solve(x=0.5,0.6,eint1(x)-x)

solve(x=0.5,0.6,-Euler()-log(x)-suminf(k=1,(-x)^k/(k*k!))-x)

A358981 Decimal expansion of Pi/3 - sqrt(3)/4.

Original entry on oeis.org

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

Offset: 0

Michal Paulovic, Dec 08 2022

The constant is the area of a circular segment bounded by an arc of 2*Pi/3 radians (120 degrees) of a unit circle and by a chord of length sqrt(3). Three such segments result when an equilateral triangle with side length sqrt(3) is circumscribed by a unit circle. The area of each segment is:
  A = (R^2 / 2) * (theta - sin(theta))
  A = (1^2 / 2) * (2*Pi/3 - sin(2*Pi/3))
  A = (1 / 2) * (2*Pi/3 - sqrt(3)/2)
  A = Pi/3 - sqrt(3)/4 = (Pi - 3*sqrt(3)/4) / 3 = 0.61418484...
where Pi (A000796) is the area of the circle, and 3*sqrt(3)/4 (A104954) is the area of the inscribed equilateral triangle.
The sagitta (height) of the circular segment is:
  h = R * (1 - cos(theta/2))
  h = 1 * (1 - cos(Pi/3))
  h = 1 - 1/2 = 0.5 (A020761)

			0.6141848493043784...

Index entries for transcendental numbers

Cf. A000796, A019670, A020761, A093731, A104954, A120011.

Maple
```
evalf(Pi/3-sqrt(3)/4);
```

RealDigits[Pi/3 - Sqrt[3]/4, 10, 100][[1]]

PARI
```
Pi/3 - sqrt(3)/4
```

Equals A019670 - A120011. - Omar E. Pol, Dec 08 2022

Equals A093731 / 2. - Michal Paulovic, Mar 08 2024

A358148 Aliquot sequence starting at 326.

Original entry on oeis.org

326, 166, 86, 46, 26, 16, 15, 9, 4, 3, 1, 0

Offset: 0

Michal Paulovic, Oct 31 2022

Starting with untouchable number 326, this sequence is not a part of any larger aliquot sequence.

The sequence's pattern: 2^5*10+6, 2^4*10+6, 2^3*10+6, 2^2*10+6, 2^1*10+6, 2^0*10+6, 2^4-1, 2^3+1, 2^2-0, 2^1+1, 2^0-1.

Includes A143759.

with(numtheory): 326; while % > 0 do sigma(%)-% end do;

NestList[If[# == 0, 0, DivisorSigma[1, #] - #] &, 326, 11]

x=326; print1(x ", "); while(x, x=sigma(x)-x; if(x, print1(x ", "), print1(x)))

a(n+1) = A001065(a(n)).
G.f.: 326 + 166*x + 86*x^2 + 46*x^3 + 26*x^4 + 16*x^5 + 15*x^6 + 9*x^7 + 4*x^8 + 3*x^9 + x^10.
E.g.f.: (1182988800 + 602380800*x + 156038400*x^2 + 27820800*x^3 + 3931200*x^4 + 483840*x^5 + 75600*x^6 + 6480*x^7 + 360*x^8 + 30*x^9 + x^10)/3628800.

A357715 Decimal expansion of sqrt(16 + 32 / sqrt(5)).

Original entry on oeis.org

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

Offset: 1

Michal Paulovic, Oct 10 2022

The perimeter of a golden rectangle inscribed in a unit circle.
The width and height of the rectangle are:
  W = sqrt(2 - 2 / sqrt(5)) = A179290.
  H = sqrt(2 + 2 / sqrt(5)) = A121570.

			5.5055276818846941...

Cf. A019934, A019952, A019970, A121570, A179290, A204188, A356869.

Maple
```
sqrt(16 + 32 / sqrt(5));
```
Mathematica
```
Sqrt[16 + 32/Sqrt[5]]
```
PARI
```
sqrt(16 + 32 / sqrt(5))
```

Equals (4 / sqrt(5)) * sqrt(5 + 2 * sqrt(5)) = A356869 * A019970.
Equals sqrt(5 + 2 * sqrt(5)) / (sqrt(5) / 4) = A019970 / A204188.
Equals 4 * sqrt(1 + 2 / sqrt(5)) = 4 * A019952.
Equals 4 / sqrt(5 - 2 * sqrt(5)) = 4 / A019934.

A356869 Decimal expansion of 4 / sqrt(5).

Original entry on oeis.org

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

Offset: 1

Michal Paulovic, Sep 01 2022

The area of a golden rectangle inscribed in a unit circle.
The width and height of the rectangle are:
  W = sqrt(2 - 2 / sqrt(5)) = A179290.
  H = sqrt(2 + 2 / sqrt(5)) = A121570.

			1.7888543819998317...

Cf. A121570, A179290, A204188.

cell2mat(struct2cell(struct(vpa(4 / sqrt(5), 105)))); ans(1:98)

parse(substring(convert(evalf(4 / sqrt(5), 105), string), 1..98));

RealDigits[4 / Sqrt[5], 10, 105][[1]][[Range[1, 97]]]

Equals [1; 1, 3, 1, 2] (periodic continued fraction expansion). - Peter Luschny, Sep 02 2022

Equals 1/A204188. - Alois P. Heinz, Sep 02 2022

A342977 Decimal expansion of (Pi - 2) / 4.

Original entry on oeis.org

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

Offset: 0

Michal Paulovic, Apr 01 2021

The constant represents the area of a circular segment bounded by an arc of Pi/2 radians (the right angle) of a unit circle and by a chord of the length of sqrt(2). Four such segments result when a square with the side length of sqrt(2) is circumscribed by a unit circle. The area of each segment is:
  A = (R^2 / 2) * (theta - sin(theta))
  A = (1^2 / 2) * (Pi/2 - sin(Pi/2))
  A = (1 / 2) * (Pi/2 - 1)
  A = (Pi - 2) / 4 = 0.28539816...
  where Pi = 3.14159265... (A000796) is the area bounded by the unit circle, and 2 is the area of the inscribed square.
Apart from the first digit this is the same as Pi/4 = 0.78539816... (A003881), the area of a circular sector bounded by the arc of Pi/2 = 1.57079632... (A019669) radians of the unit circle and by two radii of unit length, and 1/2 = 0.5 (A020761) is one-quarter of the area of the inscribed square.
The constant is close to 2/7 = 0.28571428... (2 * A020806) and Pi/11 = 0.28559933... (A019678). The equation (x - 2)/4 = x/11 has a solution x = 22/7 = 3.14285714... (A068028), which is an approximation of Pi.
The best rational approximation of the constant using small positive integers (less than 1000) is 129/452 = 0.28539823..., the next best approximation is 4771/16717 = 0.28539809...
The reciprocal of the constant is:
  1/A = 4 / (Pi - 2) = 3.50387678... (A309091).
The sagitta (height) of the circular segment is:
  h = R * (1 - cos(theta/2))
  h = 1 * (1 - cos(Pi/4))
  h = 1 - sqrt(2) / 2
  h = 1 - 1 / sqrt(2) = 0.29289321... (A268682).

			0.2853981633974483...

Index entries for transcendental numbers

Cf. A000796, A019669, A019678, A020761, A020806, A068028, A268682, A309091. Essentially the same as A003881.

RealDigits[Pi/4 - 1/2, 10, 100][[1]] (* Amiram Eldar, Jun 08 2021 *)

PARI
```
(Pi - 2) / 4
```

Equals Integral_{x=-sqrt(2)/2..sqrt(2)/2} Integral_{y=sqrt(2)/2..sqrt(1-x^2)} dy dx.
Equals Sum_{k>=1} (-1)^(k + 1)/(4*k^2 - 1). - Amiram Eldar, Jun 08 2021
Continued fraction: 1/(3 + 3/(4 + 15/(4 + 35/(4 + ... + (4*n^2 - 1)/(4 + ...). - Peter Bala, Feb 22 2024

A342355 Decimal expansion of 163/log(163).

Original entry on oeis.org

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

Offset: 2

Michal Paulovic, Mar 08 2021

A near-integer close to 32.

			31.9999987384900826...

MathOverflow, Why Is 163/ln(163) a Near-Integer?
MathOverflow, When is n/ln(n) close to an integer?
Index entries for transcendental numbers

Cf. A050499, A131920, A178805.

MATLAB
```
format long; 163 / log(163)
```
Maple
```
Digits:=100; evalf(163/ln(163));
```
Mathematica
```
RealDigits[163/Log[163], 10, 100][[1]]
```

default(realprecision, 100); 163 / log(163)

Equals 163/log(163).

cp's OEIS Frontend

User: Michal Paulovic

A370393 Decimal expansion of the area of a unit heptadecagon (17-gon).

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A364966 Decimal expansion of the solution to exp(-x^2) = x.

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A364931 Decimal expansion of the solution to 1 / (1 + e^(-x)) = x.

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A364521 Decimal expansion of the solution to Ei(x) = x.

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PARI

A358981 Decimal expansion of Pi/3 - sqrt(3)/4.

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A358148 Aliquot sequence starting at 326.

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A357715 Decimal expansion of sqrt(16 + 32 / sqrt(5)).

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