A091723 -id:A091723 - cp's OEIS frontend

A228507 Numbers n such that first n digits of A091723 (Decimal expansion of the root x of ExpIntegralEi[x]==0) gives a prime number.

Original entry on oeis.org

1, 2, 145, 180, 691, 4123

Offset: 1

Robert Price, Aug 23 2013

a(7) > 10^4.

Eric Weisstein's World of Mathematics, Constant Primes
Eric Weisstein's World of Mathematics, Integer Sequence Primes

Cf. A091723.

A070769 Decimal expansion of Soldner's constant.

Original entry on oeis.org

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

Offset: 1

Eric W. Weisstein, May 05 2002

From Amiram Eldar, Aug 14 2020: (Start)
The only positive solution to li(x) = 0, where li is the logarithmic integral.
Named after the German physicist, mathematician and astronomer Johann Georg von Soldner (1776 - 1833).
Also known as Ramanujan-Soldner constant.
Mascheroni (1792) calculated the value 1.45137. Soldner (1809) calculated the value 1.4513692346. (End)

			1.45136923488338105028396848589...

Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003. See p. 425.

Robert Price, Table of n, a(n) for n = 1..10000
Bruce C. Berndt and Ronald J. Evans, Some elegant approximations and asymptotic formulas of Ramanujan, Journal of computational and applied mathematics, Vol. 37 No. 1-3 (1991), pp. 35-41. See p. 38.
Lorenzo Mascheroni, Adnotationes ad calculum integralem Euleri, In quibus nonnulla Problemata ab Eulero proposita resolvuntur, Pars altera, Petrus Galeatius, Ticini 1792. See p. 17.
Niels Nielsen, Die Gammafunktion, New York : Chelsea, 1965.
Johann Georg von Soldner, Théorie et tables d'une nouvelle fonction transcendante, München: Lindauer, 1809. See p. 42.
Eric Weisstein's World of Mathematics, Soldner's Constant.
Eric Weisstein's World of Mathematics, Logarithmic Integral.
Wikipedia, Logarithmic integral function.
Wikipedia, Ramanujan-Soldner constant.
Marek Wolf, The relations between Euler-Mascheroni and Ramanujan-Soldner constants, 2019.

Cf. A091723.

RealDigits[ x /. FindRoot[ LogIntegral[x] == 0, {x, 2}, WorkingPrecision -> 105]][[1]] (* Jean-François Alcover, Nov 08 2012 *)

solve(x=1.4,2,real(eint1(-log(x)))) \\ Charles R Greathouse IV, Feb 23 2017

Equals exp(A091723). - Amiram Eldar, Aug 14 2020

Offset corrected and example added by Stanislav Sykora, May 18 2012

A276709 Decimal expansion of the derivative of logarithmic integral at its positive real root.

Original entry on oeis.org

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

Offset: 1

Stanislav Sykora, Sep 15 2016

Since the real root location of li(x) is the Soldner's constant A070769, this constant equals 1/log(A070769). It is also the inverse of the unique real root A091723 of the exponential integral function Ei(x).

			2.68451035082070765250238264048723868531017973459855163522041486450...

Stanislav Sykora, Table of n, a(n) for n = 1..2000
Eric Weisstein's World of Mathematics, Logarithmic Integral
Wikipedia, Logarithmic integral function

Cf. A070769, A091723, A257821.

1/x/.FindRoot[ExpIntegralEi[x] == 0, {x, 1}, WorkingPrecision -> 104] (* Vaclav Kotesovec, Sep 27 2016 *)

li(z) = {my(c=z+0.0*I); \\ Computes li(z) for any complex z
if(imag(c)<0,return(-Pi*I-eint1(-log(c))),return(+Pi*I-eint1(-log(c))));}
a = 1/log(solve(x=1.1,2.0,real(li(x)))) \\ Computes this constant

Equals 1/log(A070769) and 1/A091723.

A380270 Decimal expansion of Integral_{x=1..A070769} li(x) dx (negated), where li(x) is the logarithmic integral.

Original entry on oeis.org

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

Offset: 0

Artur Jasinski, Jan 18 2025

A070769 is Soldner's constant, where li(A070769)=0.

Integral_{x=0..1} li(x) dx = -log(2) then Integral_{x=0..A070769} li(x) dx = A380270 - log(2) = -1.19324951683011591582919049...

			-0.500102336270170606411958373..

Cf. A069284, A070769, A084945, A091723, A244067, A257817, A257818, A257819, A257820, A257821, A270857.

y = x /. FindRoot[LogIntegral[x] == 0, {x, 1.5}, WorkingPrecision -> 200]; yy = -Integrate[LogIntegral[x], {x, 1, y}]; RealDigits[yy, 10, 105][[1]]

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)

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A228507 Numbers n such that first n digits of A091723 (Decimal expansion of the root x of ExpIntegralEi[x]==0) gives a prime number.

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A070769 Decimal expansion of Soldner's constant.

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A276709 Decimal expansion of the derivative of logarithmic integral at its positive real root.

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A380270 Decimal expansion of Integral_{x=1..A070769} li(x) dx (negated), where li(x) is the logarithmic integral.

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

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