A194567 -id:A194567 - cp's OEIS frontend

A256500 Decimal expansion of the positive solution to x = 2*(1-exp(-x)).

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

Offset: 1

Stanislav Sykora, Mar 31 2015

Each of the positive solutions to x = q*(1-exp(-x)) obtained for q = 2, 3, 4, and 5, appears in several formulas pertinent to Planck's black-body radiation law. For a given q, the solution can be also written as q+W(-q/exp(q)), where W is the Lambert function. Here q = 2.

The constant appears in asymptotic formula for A007820. - Vladimir Reshetnikov, Oct 10 2016

			1.5936242600400400923230418758751602417890024248188593649995...

Stanislav Sykora, Table of n, a(n) for n = 1..2000
V. Kotesovec, Non-attacking chess pieces, 6ed, 2013, p. 249.
SpectralCalc, Calculation of Blackbody Radiance, Appendix C.
Wikipedia, Planck's law
Index entries for transcendental numbers

Cf. A194567 (q=3), A256501 (q=4), A256502 (q=5).

Cf. A106533, A191236, A217905, A229553.

RealDigits[2 + LambertW[-2 Exp[-2]], 10, 100][[1]] (* Vladimir Reshetnikov, Oct 10 2016 *)

a2=solve(x=0.1,10,x-2*(1-exp(-x))) \\ Use real precision in excess

Equals 2*(1-A106533). - Miko Labalan, Dec 18 2024

Equals log(A229553). - Hugo Pfoertner, Dec 19 2024

A094090 Decimal expansion of positive solution to 5(1-exp(u)) + uexp(u) = 0.

Original entry on oeis.org

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

Offset: 1

Jeppe Stig Nielsen, May 01 2004

This purely mathematical constant turns up when in physics one derives Wien's displacement law from the Planck black-body radiation law (see link).

Positive solution to x = 5*(1-exp(-x)). More comments in A256500. - Stanislav Sykora, Apr 01 2015

			u=4.965114231744276...

Stanislav Sykora, Table of n, a(n) for n = 1..2000
NIST, Wien displacement law constant, in Fundamental Physical Constants.
Eric Weisstein's World of Physics, Wien's Displacement Law
Wikipedia, Planck's law
Index entries for transcendental numbers

Cf. A194567, A256500, A256501.

RealDigits[5 + ProductLog[ -5/E^5], 10, 120][[1]] (* Robert G. Wilson v, May 04 2004 *)

a5=solve(x=0.1, 10, x-5*(1-exp(-x))) \\ Use real precision in excess

u = 5 + W(-5*exp(-5)), where W() is Lambert's W-function.

More terms from Robert G. Wilson v, May 04 2004

A256501 Decimal expansion of the positive solution to x = 4*(1-exp(-x)).

Original entry on oeis.org

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

Offset: 1

Stanislav Sykora, Apr 01 2015

Each of the positive solutions to x = q*(1-exp(-x)) obtained for q = 2, 3, 4, and 5, appears in several formulas pertinent to Planck's black-body radiation law. For a given q, the solution can be also written as q+W(-q/exp(q)), where W is the Lambert function. Here q = 4.

			3.9206903948728863435608913526135362205256273712079845304011750...

Stanislav Sykora, Table of n, a(n) for n = 1..2000
SpectralCalc, Calculation of Blackbody Radiance, Appendix C.
Wikipedia, Planck's law
Index entries for transcendental numbers

Cf. A094090 (q=5), A194567 (q=3), A256500 (q=2).

RealDigits[x/.FindRoot[x==4(1-Exp[-x]),{x,3},WorkingPrecision->120]] [[1]] (* Harvey P. Dale, May 08 2017 *)

a4=solve(x=0.1, 10, x-4*(1-exp(-x))) \\ Use real precision in excess

A345411 Decimal expansion of the ratio between the position of the peak of the wavelength distribution of black-body radiation and the wavelength corresponding to the position of the peak of this distribution in the frequency domain.

Original entry on oeis.org

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

Offset: 0

Amiram Eldar, Jun 18 2021

Equivalently, equals lambda_peak * nu_peak / c, where lambda_peak are nu_peak are the wavelength and frequency at which the maximum of the Planck function in the wavelength and frequency domains occurs, respectively, and c is the speed of light.

			0.56825266054974313110465933802170425871512309444119...

Sean M. Stewart and R. Barry Johnson, Blackbody Radiation: A History of Thermal Radiation Computational Aids and Numerical Methods, CRC Press, 2016, eq. (2.39), p. 46.

T. Richard Carson, Steps to the Planck Function: A Centenary Reflection, arXiv preprint astro-ph/0011219, 2000. See p. 10.
Gerhard Kramm and Nicole Mölders, Planck's blackbody radiation law: Presentation in different domains and determination of the related dimensional constants, arXiv:0901.1863 [physics.hist-ph], 2009.
Wikipedia, Black-body radiation.
Wikipedia, Planck's law.

Cf. A094090, A194567.

RealDigits[(3 + ProductLog[-3/E^3])/(5 + ProductLog[-5/E^5]), 10, 100][[1]]

Equals (3 + W(-3*exp(-3)))/(5 + W(-5*exp(-5))) = A194567/A094090, where W(x) is Lambert's W-function.

A357838 Decimal expansion of Wien frequency displacement law constant.

Original entry on oeis.org

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

Offset: 11

Lee A. Newberg, Oct 14 2022

			58789257576.4682494660613079530972169147514425888292...

NIST, Wien frequency displacement law constant
Wikipedia, Wien's displacement law.

Cf. A081819 (corresponding wavelength).

Cf. A003676, A070063, A194567.

Equals A194567 * A070063 / A003676. - Amiram Eldar, Oct 14 2022

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A256500 Decimal expansion of the positive solution to x = 2*(1-exp(-x)).

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A094090 Decimal expansion of positive solution to 5*(1-exp(u)) + u*exp(u) = 0.

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A256501 Decimal expansion of the positive solution to x = 4*(1-exp(-x)).

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A345411 Decimal expansion of the ratio between the position of the peak of the wavelength distribution of black-body radiation and the wavelength corresponding to the position of the peak of this distribution in the frequency domain.

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A357838 Decimal expansion of Wien frequency displacement law constant.

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A094090 Decimal expansion of positive solution to 5(1-exp(u)) + uexp(u) = 0.