  We study the properties of the product, which runs over the primes,
$$\mathfrak{p}_n = \prod_{s_p(n) \, \geq \, p} p \quad (n \geq 1),$$ where
$s_p(n)$ denotes the sum of the base-$p$ digits of $n$. One important property
is the fact that $\mathfrak{p}_n$ equals the denominator of the Bernoulli
polynomial $B_n(x) - B_n$, where we provide a short $p$-adic proof. Moreover,
we consider the decomposition $\mathfrak{p}_n = \mathfrak{p}_n^- \cdot
\mathfrak{p}_n^+$, where $\mathfrak{p}_n^+$ contains only those primes $p >
\sqrt{n}$. Let $\omega( \cdot )$ denote the number of prime divisors. We show
that $\omega( \mathfrak{p}_n^+ ) < \sqrt{n}$, while we raise the explicit
conjecture that $$\omega( \mathfrak{p}_n^+ ) \, \sim \, \kappa \,
\frac{\sqrt{n}}{\log n} \quad \text{as $n \to \infty$}$$ with a certain
constant $\kappa > 1$, supported by several computations.
