On a product of certain primes

Abstract

We study the properties of the product, which runs over the primes, $${\mathfrak{p}}_n=\prod _{s_p(n)\ge p}p(n\ge 1),$$ 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 $$\omega ({\mathfrak{p}}_n^{+})\sim \kappa \frac{\sqrt{n}}{\log n}\text{as n→∞}$$ with a certain constant $\kappa >1$, supported by several computations.

14 pages, 2 figures, to appear in the Journal of Number Theory

Cited by in corpus (1)

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