# Could someone explain this theorem from von Staudt?

This is an extract from Paulo Ribenboim: 13 lectures on Fermats last theorem on page 105. 'In 1845, von Staudt determined some factors of the numerator N_2k. Let 2k = k1*k2 with gcd(k1,k2)= 1 such that p|k2 if and only if p|D_2k'. Where N_2k and D_2k are the numerators and denominators of Bernoulli number B_2k. So I've actually used the result of this theorem for some other proof, but looking back at it I find it is not true. For example when 2k=74, then we can take k2=37. If we take p=37, we see that 37|k2=37 and so 37 must divide the denominator D_74 but D_74=6. Im not sure what I'm missing here. Maybe, I have misinterpreted the theorem. Could someone clear this up for me.

Could you explain how your question could be related to Sage? Otherwise, there are other platforms where such theoretical questions are welcome.