The MRB constant is defined at http://mathworld.wolfram.com/MRBConstant.html.
In looking for a faster method of calculating digits of the MRB constant, Sum[(1-)^x (x^(1/x)-1))], by the seemingly difficult method of solving minimal polynomials, I came across the following
where it seems Table[Expand[(1 + b)^x-x], {x, 1, 145}] = Table[MinimalPolynomial[x^(1/x)-1, b], {x, 1, 145}] for all x
except for "numbers of the form (kp)^p for prime p and k=1,2,3,...," OEIS A097764 :
(real = Table[MinimalPolynomial[-1 + x^(1/x), b], {x, 1, 145}]);
(guess = Table[Expand[(1 + b)^x]-x, {x, 1, 145}]);
real - guess // TableForm (*shows the equality for all but OEIS [A097764][3] *)
This equality could become very useful because as x gets large the minimal polynomial of x^(1/x)-1 becomes exceedingly difficult to compute!
Here is how this equality can be used:
Partial sum(s) of the MRB constant can be found through a sum of NSolves,
y = 1000; N[-y - Sum[b /. (NSolve[(1 + b)^n == n, b, Reals][[1]]), {n, 1, y}]]
giving a more correct result than
NSum[(-1)^n (n^(1/n) - 1), {n, 1, 1000}]
because it removes the imaginary part given by E^(I*Pi x) i.e. (-1)^x.