So in a recent question I was trying to prove that $2^n-1$ will never be a Carmichael number (Can a Mersenne number ever be a Carmichael number?), I was going to prove it true as long as a certain property about Bernoulli numbers occured. (I know realized I made a huge mistake, but anyway...)
The property I noticed was that the denominators of $B_{2^n}$ are the product of consecutive Fermat primes. (With $2^0+1 = 2$ included)
Ex: Denominators tested:
- $6 = 2*3$
- $30 = 2*3*5$
- $510 = 2*3*5*17$
- $131070 = 2*3*5*17*257$
Maybe even more specifically: the denominator of $B_{2^n}$ is the product of consecutive Fermat primes less than or equal to $2^n+1$.
