### Arithmetic Properties of Generalized Euler Numbers

The generalized Euler number E_{n|k} counts
the number of permutations of {1,2,...,n} which have a descent in
position *m* if and only if *m* is divisible by *k*.
The classical Euler numbers are the special case when *k=2*.
In this paper, we study divisibility properties of a *q*-analog of
E_{n|k}. In particular, we generalize two theorems of
Andrews and Gessel about factors of the *q*-tangent numbers.