A202357 -id:A202357 - cp's OEIS frontend

A248200 Decimal expansion of x in the solution to x^e = e^(-x), where e = exp(1). Also the smallest value of the constant c where there exists a solution to x^c = c^(-x).

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

Offset: 0

Richard R. Forberg, Dec 01 2014

At this value both sides of the equation x^e = e^(-x) equal: 0.46909728... .
Within the more general family of equations x^c  = c^(-x), the solution for x is smallest when c = e.
Let's name cmin this constant: 0.7569451...
The general equation x^c  = c^(-x) has real solutions only where c >= cmin.
At values of c in the range cmin < c < 1, there are two solutions.
At values of c < cmin, the two curves do not intersect.
At values of c =~ cmin, the two curves become essentially parallel over an extended range.
When c = cmin, x = e is the tangent point, where both sides of the equation equal exp(cmin) = 2.1317539... = cmin^(-e) = 1/0.46909728...
If the equation were x^e = e^x, the solution would be x = e.

			0.7569451064575836645840170881202415000611....

Cf. A001113, A068985, A202357.

RecurrenceTable [{a[n + 1] == N[1/Exp[a[n]]^(1/Exp[1]), 150],
  a[1] == 3/4}, a, {n, 1, 200}]
RealDigits[ E*ProductLog[1/E], 10, 111][[1]] (* Robert G. Wilson v, Jan 30 2015 *)

solve(x=0, 1, x^exp(1) - exp(-x)) \\ Michel Marcus, Dec 01 2014

exp(1)*lambertw(exp(-1)) \\ Gleb Koloskov, Aug 25 2021

From Gleb Koloskov, Aug 25 2021: (Start)
Equals e*LambertW(1/e) = A001113*LambertW(A068985) = A001113*A202357.
Equals Sum_{n>0} (-n/e)^(n-1)/n!. (End)

A375596 Decimal expansion of 1/(1 - W(1/e)), where W is the Lambert W function.

Original entry on oeis.org

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

Offset: 1

Stefano Spezia, Aug 20 2024

			1.38593327599819425386062181488251586064513...

Edgar Assing, On gaps between primes. See p. 45.

Cf. A202357.

Cf. A141606, A299617, A299618, A299619, A299632, A299633, A307951, A322166.

RealDigits[1/(1-ProductLog[1/E]),10,100][[1]]

1/(1-lambertw(1/exp(1))) \\ Michel Marcus, Aug 20 2024

Equals 1/(1 - A202357).

cp's OEIS Frontend

A248200 Decimal expansion of x in the solution to x^e = e^(-x), where e = exp(1). Also the smallest value of the constant c where there exists a solution to x^c = c^(-x).

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A375596 Decimal expansion of 1/(1 - W(1/e)), where W is the Lambert W function.

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