Power-Sum Denominators

The power sum 1n + 2n +… + xn has been of interest to mathematicians since classical times. Johann Faulhaber, Jacob Bernoulli, and others who followed expressed power sums as polynomials in x of degree n + 1 with rational coefficients. Here, we consider the denominators of these polynomials and prove some of their properties. A remarkable one is that such a denominator equals n + 1 times the squarefree product of certain primes p obeying the condition that the sum of the base-p digits of n + 1 is at least p. As an application, we derive a squarefree product formula for the denominators of the Bernoulli polynomials.