A256502 -id:A256502 - 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

A226762 Greatest k such that 1/k >= mean of {1, 1/2, 1/3, ..., 1/n}.

Original entry on oeis.org

1, 1, 1, 1, 2, 2, 2, 2, 3, 3, 3, 3, 4, 4, 4, 4, 4, 5, 5, 5, 5, 5, 6, 6, 6, 6, 6, 7, 7, 7, 7, 7, 8, 8, 8, 8, 8, 8, 9, 9, 9, 9, 9, 10, 10, 10, 10, 10, 10, 11, 11, 11, 11, 11, 11, 12, 12, 12, 12, 12, 12, 13, 13, 13, 13, 13, 13, 14, 14, 14, 14, 14, 14, 15, 15

Offset: 1

Clark Kimberling, Jun 19 2013

Largest integer not exceeding the harmonic mean of the first n numbers.

			1/4 < mean{1,1/2,1/3,...,1/9} < 1/3, so that a(9) = 3.

Clark Kimberling, Table of n, a(n) for n = 1..1000

Cf. A226763, A256502.

f[n_] := Mean[Table[1/k, {k, 1, n}]]
Table[Floor[1/f[n]], {n, 1, 120}]   (* this sequence *)
Table[Ceiling[1/f[n]], {n, 1, 120}] (* A226763 *)

\\ This uses only precision-independent integer operations:
a(n)=(n*n!)\sum(k=1,n,n!\k)  \\ Stanislav Sykora, Apr 08 2015

a(n) = floor(n/(Sum_{k=1..n} 1/k)).

Name corrected by Jason Yuen, Nov 02 2024

cp's OEIS Frontend

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

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