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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.

##### MSC:
 11B83 Special sequences and polynomials 11B68 Bernoulli and Euler numbers and polynomials
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