Show that $n+1$ is prime if (Denominator(Bernoulli Number($n$)))/($n+1$) is an integer

Question

I suspect this is true, but I'm having trouble proving it. Here are some example cases:

$$6/3=2$$ $$30/5=7$$ $$66/11=6$$ ... $$9538864545210/241=39580350810$$

and I suppose a few examples of it not being an integer for composite numbers is a good idea:

$$30/9=3.\overline{33}$$ $$6/15=.4$$ $$330/21=110/7\approx15.7143$$ ... $$30/249\approx0.120482$$

In my last post, I noticed a few extremely intriguing properties (previously known); one of which is that the polynomial solutions to the sum of the first powers of n up to k have coefficients related to the Bernoulli numbers. Another is that these coefficients (as required for these polynomials to product integers) sum to one. I have noticed quite a few more things, but I'm not sure they are relevant.

The reason I ask this is because I further suspect that this could potentially reveal some useful relationships about prime numbers, and the Bernoulli numbers. (Disclaimer: I am not a professional mathematician. If I miss something obvious, please be kind).

Answer 1

This is a theorem known as Von Staudt-Clausen that shows, as direct consequence of the nature of Bernoulli numbers, that Bernoulli fractions have unique prime denominators.

Specifically, every Bernoulli expansion has factors of (n+1) in it, and no integers other than prime integers can divide the denominator of $B_n$.