  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.
