Kellner, Bernd C. On a product of certain primes. (English) Zbl 1418.11045 J. Number Theory 179, 126-141 (2017). 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. Cited in 6 Documents MSC: 11B83 Special sequences and polynomials 11B68 Bernoulli and Euler numbers and polynomials Keywords:product of primes; Bernoulli polynomials; denominator; sum of base-\(p\) digits; \(p\)-adic valuation of polynomials PDF BibTeX XML Cite \textit{B. C. Kellner}, J. Number Theory 179, 126--141 (2017; Zbl 1418.11045) Full Text: DOI arXiv Online Encyclopedia of Integer Sequences: a(n) is the smallest positive integer such that a(n)*(1^n + 2^n + ... + x^n) is a polynomial in x with integer coefficients. Least number k such that all coefficients of k*B(n,x), the n-th Bernoulli polynomial, are integers. a(n) = denominator(Bernoulli_{n+1}(x) - Bernoulli_{n+1}). a(n) = denominator(Bernoulli_{n}(x)) / denominator(Bernoulli_{n}). a(n) = b(2*n-1)/b(2*n) where b(n) = A195441(n-1) = denominator(Bernoulli_{n}(x) - Bernoulli_{n}). a(n) = b(2*n)/b(2*n+1) where b(n) = denominator(Bernoulli_{n}(x)). Indices k such that A195441(k) = A195441(k+1). Numbers that appear in A195441 at least once for two consecutive indices. Product of all primes p dividing n such that the sum of the base p digits of n is at least p, or 1 if no such prime. Product of all primes p not dividing n such that the sum of the base p digits of n is at least p, or 1 if no such prime. Product of all primes p dividing n such that the sum of the base p digits of n is less than p, or 1 if no such prime. 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: 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: 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: 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 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.