OFFSET
1,2
COMMENTS
The least real solution of x^2 = 2^sqrt(x). This equation has two real solutions the other is 256.
Let x be this constant, and c = 2*log(x)/log(2); then c^4 = 2^c.
Let x be this constant, and c = 1/sqrt(x); then c^c = 1/2^(1/4).
FORMULA
Equals e^(-2*Sum_{k>=1} ((-k)^(-1+k)*(-log(2)/4)^k/k!)).
Equals e^(t*log(2)/2) where t = (2^(1/4))^(2^(1/4))^(2^(1/4))^(2^(1/4))^... is the infinite power tower over 2^(1/4).
Equals 16*LambertW(-log(2)/4)^2 / log(2)^2. - Vaclav Kotesovec, May 22 2023
EXAMPLE
1.5366769...
MATHEMATICA
RealDigits[E^(-2 ProductLog[-Log[2]/4]), 10, 100][[1]]
PROG
(PARI)
\p 200
exp(-2*lambertw(-log(2)/4))
(Python)
import math; from sympy import LambertW
print([i for i in str("%.30f" % math.exp(-2*LambertW(-math.log(2)/4)))])
# Javier Rivera Romeu, May 22 2023
CROSSREFS
KEYWORD
nonn,cons
AUTHOR
Thomas Scheuerle, May 22 2023
STATUS
approved