  We study the integrality properties of the coefficients of the mirror map
attached to the generalized hypergeometric function $_{n}F_{n-1}$ with rational
parameters and with a maximal unipotent monodromy. We present a conjecture on
the $p$-integrality of the mirror map which can be verified experimentally. We
prove its consequence on the $N$-integrality of the mirror map for the
particular cases $1\leq n\leq 4$.
  For $n=2$ we obtain the Takeuchi's classification of arithmetic triangle
groups with a cusp, and for $n=4$ we prove that $14$ examples of hypergeometric
Calabi-Yau equations are the full classification of hypergeometric mirror maps
with integral coefficients. As a by-product we get the integrality of the
corresponding algebra of modular-type functions. These are natural
generalizations of the algebra of classical modular and quasi-modular forms in
the case $n=2$.
