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).

%I A248200 #46 Sep 03 2021 10:58:05
%S A248200 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,
%T A248200 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,
%U A248200 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
%N 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).
%C A248200 At this value both sides of the equation x^e = e^(-x) equal: 0.46909728... .
%C A248200 Within the more general family of equations x^c  = c^(-x), the solution for x is smallest when c = e.
%C A248200 Let's name cmin this constant: 0.7569451...
%C A248200 The general equation x^c  = c^(-x) has real solutions only where c >= cmin.
%C A248200 At values of c in the range cmin < c < 1, there are two solutions.
%C A248200 At values of c < cmin, the two curves do not intersect.
%C A248200 At values of c =~ cmin, the two curves become essentially parallel over an extended range.
%C A248200 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...
%C A248200 If the equation were x^e = e^x, the solution would be x = e.
%F A248200 From _Gleb Koloskov_, Aug 25 2021: (Start)
%F A248200 Equals e*LambertW(1/e) = A001113*LambertW(A068985) = A001113*A202357.
%F A248200 Equals Sum_{n>0} (-n/e)^(n-1)/n!. (End)
%e A248200 0.7569451064575836645840170881202415000611....
%t A248200 RecurrenceTable [{a[n + 1] == N[1/Exp[a[n]]^(1/Exp[1]), 150],
%t A248200   a[1] == 3/4}, a, {n, 1, 200}]
%t A248200 RealDigits[ E*ProductLog[1/E], 10, 111][[1]] (* _Robert G. Wilson v_, Jan 30 2015 *)
%o A248200 (PARI) solve(x=0, 1, x^exp(1) - exp(-x)) \\ _Michel Marcus_, Dec 01 2014
%o A248200 (PARI) exp(1)*lambertw(exp(-1)) \\ _Gleb Koloskov_, Aug 25 2021
%Y A248200 Cf. A001113, A068985, A202357.
%K A248200 nonn,cons
%O A248200 0,1
%A A248200 _Richard R. Forberg_, Dec 01 2014