Let m and n be positive integers.
Let $f(m,0)=m$
Let $f(m,n)= e \ln(f(m,n-1))$
$\lim_{m\to\infty} \ln(m)\Big(f(m,\lfloor\ln m\rfloor)) - e\Big) = 163^{1/3}+C$
Where $C$ is a constant.
It seems $0.005 > C > 0$
Is this true ? Why is this so ? Is $C = 0$ ?
Is this an analogue to the computation of the Paris constant ?
Can we give a closed form for $C$ ?
EDIT :
Conjecture :
$\lim_{m\to\infty} \ln(m)\Big(f(m,\lfloor\ln m\rfloor)) - e\Big) = A$
$A > 0$
Is this true ? How to prove this ?