On a product of certain primes. (English) Zbl 1418.11045

Summary: We study the properties of the product, which runs over the primes, \[ \mathfrak{p}_n = \prod_{s_p(n) \geq p} p(n \geq 1), \] 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 \[ \omega(\mathfrak{p}_n^+) \sim \kappa \frac{\sqrt{n}}{\log n}\quad \text{as}\, n \rightarrow \infty \] with a certain constant \(\kappa > 1\), supported by several computations.


11B83 Special sequences and polynomials
11B68 Bernoulli and Euler numbers and polynomials
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