On a product of certain primes

We study the properties of the product, which runs over the primes, P-n = Pi(sp(n)>= p) p (n >= 1), sp 01) > P where s(p)(n) denotes the sum of the base-p digits of n. One important property is the fact that 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 P-n = P-n(-) . P-n(+), where p(n)(+) contains only those primes p > root n. Let omega(.) denote the number of prime divisors. We show that omega(p(n)(+)) < root n, while we raise the explicit conjecture that omega(P-n(+)) similar to kappa (log n)/(root n) as n -> infinity with a certain constant kappa > 1, supported by several computations. (C) 2017 Elsevier Inc. All rights reserved.