Computing the number of $sigma$-algebra.

Algebras (and thus $sigma$-algebras) of subsets of a finite set correspond to partitions of the set. If $P = {P_1, ldots, P_k}$ is a partition of $X$, i.e. disjoint nonempty subsets whose union is $X$, then you get an algebra consisting of the $2^k$ unions of subsets of $P$. So if $m = 2^k$, the number of algebras of subsets of $X$ is the number of partitions of $X$ into $k$ parts. If $m$ is not a power of $2$, there are no algebras with cardinality $m$.