  We show that the Taylor coefficients of the series ${\bf q}(z)=z\exp({\bf
G}(z)/{\bf F}(z))$ are integers, where ${\bf F}(z)$ and ${\bf G}(z)+\log(z)
{\bf F}(z)$ are specific solutions of certain hypergeometric differential
equations with maximal unipotent monodromy at $z=0$. We also address the
question of finding the largest integer $u$ such that the Taylor coefficients
of $(z ^{-1}{\bf q}(z))^{1/u}$ are still integers. As consequences, we are able
to prove numerous integrality results for the Taylor coefficients of mirror
maps of Calabi-Yau complete intersections in weighted projective spaces, which
improve and refine previous results by Lian and Yau, and by Zudilin. In
particular, we prove the general ``integrality'' conjecture of Zudilin about
these mirror maps. A further outcome of the present study is the determination
of the Dwork-Kontsevich sequence $(u_N)_{N\ge1}$, where $u_N$ is the largest
integer such that $q(z)^{1/u_N}$ is a series with integer coefficients, where
$q(z)=\exp(F(z)/G(z))$, $F(z)=\sum_{m=0} ^{\infty} (Nm)! z^m/m!^N$ and
$G(z)=\sum_{m=1} ^{\infty} (H_{Nm}-H_m)(Nm)! z^m/m!^N$, with $H_n$ denoting the
$n$-th harmonic number, conditional on the conjecture that there are no prime
number $p$ and integer $N$ such that the $p$-adic valuation of $H_N-1$ is
strictly greater than 3.
