Further divisibility properties of the $q$-tangent numbers

Abstract:

The $q$-tangent number ${T_{2n + 1}}(q)$ is shown to be divisible by ${(1 + q)^{a(n,1)}}{(1 + {q^2})^{a(n,2)}} \cdots {(1 + {q^n})^{a(n,n)}}$, where the $a(n,i)$’s are positive integers having the maximal property that $a(n,1) + a(n,2) + \cdots + a(n,n) = 2n$ whenever $n$ is a power of 2.