Arithmetical turbulence of selfsimilar fluctuations statistics of large Frobenius numbers of additive semigroups of integers

The Frobenius number of vector $a$, whose components as are natural numbers (having no common divisor greater than 1), is the minimal integer $N(a)$ which is representable as a sum of the components as with nonnegative multiplicities, together with all the greater integers (like for $N(4,5)=12$).
The mean Frobenius number is the arithmetical mean value of the number $N(a)$ along the simplex of the vectors a for which $a_1+\dots+a_n=\sigma$.
Numerical experiments suggest the growth rate of this mean value (for large $\sigma$) of order $\sigma у^p$, where $p=1+1/(n-1)$, that is of order $3/2$ for $N(a,b,c)$.
Fluctuations are making some of the Frobenius numbers many times higher at the place of some resonances, like $b=c$.
The selfsimilar statistics of the fluctuations, contained in the present article, suggest, that these fluctuations are insufficiently frequent to influence the behaviour of the mean value at large scales $\sigma$.