  We study an integer sequence associated with Cantor's division polynomials of
a genus $2$ curve having an integral point. We show that the modulo $p$ of such
a sequence is periodic for all but finitely many primes $p$, and describe the
relation between the period of the modulo $p$ of the sequence and the order of
the integral point on the reduction modulo $p$ in the Jacobian variety
explicitly. This generalizes Ward's results on elliptic divisibility sequences
associated with division polynomials of elliptic curves.
