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.
Categories: math.NT