## 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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### References:

 [1] Carlitz, L., A divisibility property of the binomial coefficients, Amer. Math. Monthly, 68, 560-561, (1961) · Zbl 0113.26602 [2] Cohen, H., Number theory, volume II: analytic and modern tools, Grad. Texts in Math., vol. 240, (2007), Springer-Verlag New York · Zbl 1119.11002 [3] Kellner, B. C.; Sondow, J., Power-sum denominators, Amer. Math. Monthly, (2017/2018), in press · Zbl 1391.11052 [4] Prasolov, V. V., Polynomials, Algorithms Comput. Math., vol. 11, (2010), Springer-Verlag Berlin · Zbl 1272.12001 [5] Robert, A. M., A course in p-adic analysis, Grad. Texts in Math., vol. 198, (2000), Springer-Verlag New York · Zbl 0947.11035 [6] Rosser, J.; Schoenfeld, L., Approximate formulas for some functions of prime numbers, Illinois J. Math., 6, 64-94, (1962) · Zbl 0122.05001
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