Faulhaber's formula, Bernoulli numbers, power sums of natural numbers and totatives and the functional equation $f(x)+x^k=f(x+1)$

Abstract

In modern usage the Bernoulli numbers and Bernoulli polynomials follow Euler's approach and are defined using generating functions. Originally, they were derived by Bernoulli while characterizing Faulhaber's formula for the sum of consecutive powers. These equations have many consequences and applications in various areas of mathematics. We consider yet another application by studying the functional equation $f(x)+x^k=f(x+1)$ and show that a solution of this equation can be derived from Faulhaber's formula. We then use these results to study the totatives of n, i.e. numbers less than n that are coprime to n. In particular, we look at sums of powers of totatives of n that are less than n/2. We show that the sum of powers of this half of the totatives can also be expressed in the same structural form as the sum of powers of all totatives and provide explicit formulas for this sum. As an application of these results, we obtain a formula for the total area of all rectangles with coprime width and length and semiperimeter n.