## Power-sum denominators.(English)Zbl 1391.11052

Summary: The power sum $$1^n+2^n+\dots+x^n$$ 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.

### MSC:

 11B83 Special sequences and polynomials 11B68 Bernoulli and Euler numbers and polynomials

### Keywords:

Bernoulli polynomials

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