We demonstrate how most common cardinal invariants associated with a von Neumann algebra $\mathcal M$ can be computed from the decomposability number, $\operatorname{dens}(\mathcal M)$, and the minimal cardinality of a generating set, $\operatorname{gen}(\mathcal M)$. Applications include the equivalence of the well-known generator problem, ``Is every separably-acting von Neumann algebra singly-generated?", with the formally stronger questions, ``Is every countably-generated von Neumann algebra singly-generated?" and ``Is the $\operatorname{gen}$ invariant monotone?" Modulo the generator problem, we determine the range of the invariant $\bigl( \operatorname{gen}(\mathcal M), \operatorname{dens}(\mathcal M) \bigr)$, which is mostly governed by the inequality $\operatorname{dens}(\mathcal M) \leq \mathfrak C^{\operatorname{gen}(\mathcal M)}$.

Let $a$ be a natural number greater than $1$. Let $f_a(n)$ be the order of $a$ mod $n$. Denote by $\omega(n)$ the number of distinct prime factors of $n$. Assuming a weak form of the generalised Riemann hypothesis, we prove the following conjecture of Erd\"os and Pomerance: The number of $n\leq x$ coprime to $a$ satisfying $$\alpha \leq \frac{\omega(f_a(n)) - (\log \log n)^2/2 }{ (\log \log n)^{3/2}/\sqrt{3}} \leq \beta $$ is asymptotic to $$\left(\frac{ 1 }{ \sqrt{2\pi}} \int_{\alpha}^{\beta} e^{-t^2/2}dt\right) \frac{x\phi(a) }{ a}, $$ as $x$ tends to infinity.