The paper studies the $K$-theoretic invariants of the crossed product $C^{*}$-algebras associated with an important family of homeomorphisms of the tori $\mathbb{T}^{n}$ called Furstenberg transformations. Using the Pimsner-Voiculescu theorem, we prove that given $n$, the $K$-groups of those crossed products, whose corresponding $n\times n$ integer matrices are unipotent of maximal degree, always have the same rank $a_{n}$. We show using the theory developed here that a claim made in the literature about the torsion subgroups of these $K$-groups is false. Using the representation theory of the simple Lie algebra $\frak{sl}(2,\mathbb{C})$, we show that, remarkably, $a_{n}$ has a combinatorial significance. For example, every $a_{2n+1}$ is just the number of ways that $0$ can be represented as a sum of integers between $-n$ and $n$ (with no repetitions). By adapting an argument of van Lint (in which he answered a question of Erd\H{o}s), a simple, explicit formula for the asymptotic behavior of the sequence $\{a_{n}\}$ is given. Finally, we describe the order structure of the $K_{0}$-groups of an important class of Furstenberg crossed products, obtaining their complete Elliott invariant using classification results of H. Lin and N. C. Phillips.

In this paper, we extend the approach first outlined by Hardy and Ramanujan for calculating the asymptotic formulae for the number of partitions into $r$-th powers of primes, $p_{\mathbb{P}^{(r)}}(n)$, to include their difference functions. In doing so, we rectify an oversight of said authors, namely that the first difference function is perforce positive for all values of $n$, and include the magnitude of the error term.