  For a positive integer $n$ let $\mathfrak{P}_n=\prod_{s_p(n)\ge p} p,$ where
$p$ runs over all primes and $s_p(n)$ is the sum of the base $p$ digits of $n$.
For all $n$ we prove that $\mathfrak{P}_n$ is divisible by all "small" primes
with at most one exception. We also show that $\mathfrak{P}_n$ is large, has
many prime factors exceeding $\sqrt{n}$, with the largest one exceeding
$n^{20/37}$. We establish Kellner's conjecture, which says that the number of
prime factors exceeding $\sqrt{n}$ grows asymptotically as $\kappa
\sqrt{n}/\log n$ for some constant $\kappa$ with $\kappa=2$. Further, we
compare the sizes of $\mathfrak{P}_n$ and $\mathfrak{P}_{n+1}$, leading to the
somewhat surprising conclusion that although $\mathfrak{P}_n$ tends to infinity
with $n$, the inequality $\mathfrak{P}_n>\mathfrak{P}_{n+1}$ is more frequent
than its reverse.
