A note on the denominators of Bernoulli numbers (Q464682)

Using the von Staudt-Clausen theorem the authors prove an interesting formula for the denominator of the Bernoulli numbers \(B_m\) involving the Stirling numbers of the second kind \(S(k,n)\).NEWLINENEWLINEPut \(\tilde S(k,n)=k!S(k,n)\) then the formula NEWLINE\[NEWLINE\gcd(\tilde S(2n+1,2), \ldots, \tilde S(2n+1,2n+1)) = \text{denominator of}\;B_{2n}NEWLINE\]NEWLINE holds.