The Bernoulli numbers are defined as coefficients $\beta_i$ in the expansion
$\frac{x}{e^x-1}=\beta_0 + \frac{\beta_1x}{1!} + \frac{\beta_2x^2}{2!} + \frac{\beta_3x^3}{3!} + .....$
where $\beta_0=1$,$\beta_1=-\frac{1}{2}$ and for $i \ge 1$ $\beta_{2i+1}=0$.
Hardy and wright in their book first state the following theorem by Von Staudt regarding Bernoulli numbers
$\beta_k + \sum_{p-1|k} \frac{1}{p} = l$ , where $l$ is an integer and summation is over primes $p$ such that $p-1$ divides k.
Right after stating the theorem the authors say the following
" In particular Von Staudt's theorem shows there that there is no squared factor in the denominator of any Bernoullian number ".
What do the authors mean by this ? When they say "denominator of any Bernoullian number" , do they consider the Bernoulli numbers in reduced form ? What am I missing ?