On a product of certain primes

We study the properties of the product, which runs over the primes,pn=∏sp(n)≥pp(n≥1), where sp(n) denotes the sum of the base-p digits of n. One important property is the fact that pn equals the denominator of the Bernoulli polynomial Bn(x)−Bn, where we provide a short p-adic proof. Moreover, we consider the decomposition pn=pn−⋅pn+, where pn+ contains only those primes p>n. Let ω(⋅) denote the number of prime divisors. We show that ω(pn+)1, supported by several computations.