Hello guys! Firstly, I'd like to start with a problem: Find all natural numbers n, for which n divides (10^n)1. Then, I wanna generalize the statement as: Is it possible to find all solutions to the the statement: n divides (a^n)1, where g.c.d(a,n)=1? Is there any method or algorithm at least, in this case?

The question for $a=10$ was investigated in
C. Smyth has a paper "The terms in Lucas sequences divisible by their indices" where a similar procedure is described for the general case $na^nb^n$. The main theorem is as follows:
This allows you to search recursively for allowed primes, and I doubt that there is a more elegant method for generating such numbers. 


I would just add a short sentence to the above answer. Since you are only interested in the case when $b=1$ there is a very simple elementary characterization: $na^n1$ if and only if $\mathrm{ord}_n(a)n$. That also explains the comment above that $3^m10^{3^m}1$. Unfortunately I think it also means that there is no good `formula' for the answer to your problem. 

